| L(s) = 1 | + 4·4-s + 6·16-s + 32·25-s − 32·37-s + 64·43-s − 32·67-s + 128·100-s + 32·109-s − 24·121-s + 127-s + 131-s + 137-s + 139-s − 128·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 64·169-s + 256·172-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | + 2·4-s + 3/2·16-s + 32/5·25-s − 5.26·37-s + 9.75·43-s − 3.90·67-s + 64/5·100-s + 3.06·109-s − 2.18·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 10.5·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 4.92·169-s + 19.5·172-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.236806845\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.236806845\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( ( 1 - 16 T^{2} + 144 T^{4} - 992 T^{6} + 5519 T^{8} - 992 p^{2} T^{10} + 144 p^{4} T^{12} - 16 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 11 | \( ( 1 + 6 T^{2} - 85 T^{4} + 6 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 13 | \( ( 1 + 16 T^{2} + 304 T^{4} + 16 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 17 | \( ( 1 - 32 T^{2} + 352 T^{4} - 3008 T^{6} + 72127 T^{8} - 3008 p^{2} T^{10} + 352 p^{4} T^{12} - 32 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 19 | \( ( 1 + 12 T^{2} - 102 T^{4} - 5712 T^{6} - 121789 T^{8} - 5712 p^{2} T^{10} - 102 p^{4} T^{12} + 12 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 23 | \( ( 1 + 14 T^{2} - 333 T^{4} + 14 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 29 | \( ( 1 - 20 T^{2} - 266 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 31 | \( ( 1 - 4 T^{2} + 138 T^{4} + 8176 T^{6} - 924013 T^{8} + 8176 p^{2} T^{10} + 138 p^{4} T^{12} - 4 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 37 | \( ( 1 + 8 T - 8 T^{2} - 16 T^{3} + 1447 T^{4} - 16 p T^{5} - 8 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 41 | \( ( 1 + 160 T^{2} + 9760 T^{4} + 160 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 43 | \( ( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{8} \) |
| 47 | \( ( 1 - 124 T^{2} + 7626 T^{4} - 413168 T^{6} + 21217235 T^{8} - 413168 p^{2} T^{10} + 7626 p^{4} T^{12} - 124 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 53 | \( ( 1 + 144 T^{2} + 11086 T^{4} + 580608 T^{6} + 26984595 T^{8} + 580608 p^{2} T^{10} + 11086 p^{4} T^{12} + 144 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 59 | \( ( 1 - 108 T^{2} + 3834 T^{4} - 93744 T^{6} + 8589155 T^{8} - 93744 p^{2} T^{10} + 3834 p^{4} T^{12} - 108 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 61 | \( ( 1 - 16 T^{2} - 2832 T^{4} + 69664 T^{6} - 5262673 T^{8} + 69664 p^{2} T^{10} - 2832 p^{4} T^{12} - 16 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 67 | \( ( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{8} \) |
| 71 | \( ( 1 - 92 T^{2} + 4006 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 73 | \( ( 1 + 192 T^{2} + 18240 T^{4} + 1529472 T^{6} + 122006879 T^{8} + 1529472 p^{2} T^{10} + 18240 p^{4} T^{12} + 192 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 79 | \( ( 1 - 126 T^{2} + 9635 T^{4} - 126 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 83 | \( ( 1 + 12 T^{2} + 13302 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 89 | \( ( 1 + 12480 T^{4} + 93008159 T^{8} + 12480 p^{4} T^{12} + p^{8} T^{16} )^{2} \) |
| 97 | \( ( 1 - 288 T^{2} + 38304 T^{4} - 288 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.59476526373071322015650359172, −2.58701987378957062080779455486, −2.56665972151337115144916691481, −2.49120526454418831336126461183, −2.46768076279865992675536353862, −2.36715720676269830969000801158, −2.28844164413431453363817883953, −2.14975041417111786543715111333, −1.94588072961817798478820394156, −1.86634302673386303453279920826, −1.82951390288161574977496844553, −1.67104344085099771139611516747, −1.66000212433659484320019330533, −1.47305252391249901357811057480, −1.40719838172047177416801240357, −1.38500221611655612635504749034, −1.25236320325932706255063459370, −1.17820585585358548547524936475, −1.09300852239884560333977421185, −0.950714021801436046463390408505, −0.807720520928567578003421288839, −0.66390124702045909046297591474, −0.49075837681639191947629104063, −0.47454238874623795472017816564, −0.04877627001747407857551473574,
0.04877627001747407857551473574, 0.47454238874623795472017816564, 0.49075837681639191947629104063, 0.66390124702045909046297591474, 0.807720520928567578003421288839, 0.950714021801436046463390408505, 1.09300852239884560333977421185, 1.17820585585358548547524936475, 1.25236320325932706255063459370, 1.38500221611655612635504749034, 1.40719838172047177416801240357, 1.47305252391249901357811057480, 1.66000212433659484320019330533, 1.67104344085099771139611516747, 1.82951390288161574977496844553, 1.86634302673386303453279920826, 1.94588072961817798478820394156, 2.14975041417111786543715111333, 2.28844164413431453363817883953, 2.36715720676269830969000801158, 2.46768076279865992675536353862, 2.49120526454418831336126461183, 2.56665972151337115144916691481, 2.58701987378957062080779455486, 2.59476526373071322015650359172
Plot not available for L-functions of degree greater than 10.
