| L(s) = 1 | + 8·16-s − 28·31-s + 4·61-s + 9·81-s − 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯ |
| L(s) = 1 | + 2·16-s − 5.02·31-s + 0.512·61-s + 81-s − 4·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-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 + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + 0.0646·239-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.826520497\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.826520497\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 94 T^{4} + p^{4} T^{8} \) |
| good | 2 | \( ( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} )^{2}( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 + 146 T^{4} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - p^{2} T^{4} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 37 T^{2} + p^{2} T^{4} )^{2}( 1 + 11 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | \( ( 1 - p^{2} T^{4} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + p T^{2} )^{8} \) |
| 31 | \( ( 1 - 4 T + p T^{2} )^{4}( 1 + 11 T + p T^{2} )^{4} \) |
| 37 | \( ( 1 - 2062 T^{4} + p^{4} T^{8} )( 1 + 2591 T^{4} + p^{4} T^{8} ) \) |
| 41 | \( ( 1 - p T^{2} )^{8} \) |
| 43 | \( ( 1 + 23 T^{4} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 - p^{2} T^{4} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - p^{2} T^{4} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 - 14 T + p T^{2} )^{4}( 1 + 13 T + p T^{2} )^{4} \) |
| 67 | \( ( 1 - 8809 T^{4} + p^{4} T^{8} )( 1 + 2903 T^{4} + p^{4} T^{8} ) \) |
| 71 | \( ( 1 - p T^{2} )^{8} \) |
| 73 | \( ( 1 - 8542 T^{4} + p^{4} T^{8} )( 1 - 1249 T^{4} + p^{4} T^{8} ) \) |
| 79 | \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{2}( 1 + 11 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 89 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 + 9071 T^{4} + p^{4} T^{8} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.96809413726649574342654001295, −4.80528131891132812416345878566, −4.25310315379480467698331858439, −4.24389164228588813356651247667, −4.05567267908212512396076764298, −4.00691147074192089685643714493, −3.88890130540430168255199223667, −3.87140829064623538776533328568, −3.77188898733229477127328568378, −3.48839827825580659420705717326, −3.25177775853202690774510881919, −3.13923667701264943910482808242, −2.97219976236701242639700440210, −2.91069122966161678284892715467, −2.70581375847114227469373586808, −2.51319911975242164415764382783, −2.32917428890465986019531632586, −1.86595721454693745931441196541, −1.81513170503554602224230357378, −1.74101269408170442835444652343, −1.56807672724300909126568975963, −1.46472020393295575028365369147, −0.866404681991786976776496362435, −0.66391307169393212550876107013, −0.33351506282151773829195044777,
0.33351506282151773829195044777, 0.66391307169393212550876107013, 0.866404681991786976776496362435, 1.46472020393295575028365369147, 1.56807672724300909126568975963, 1.74101269408170442835444652343, 1.81513170503554602224230357378, 1.86595721454693745931441196541, 2.32917428890465986019531632586, 2.51319911975242164415764382783, 2.70581375847114227469373586808, 2.91069122966161678284892715467, 2.97219976236701242639700440210, 3.13923667701264943910482808242, 3.25177775853202690774510881919, 3.48839827825580659420705717326, 3.77188898733229477127328568378, 3.87140829064623538776533328568, 3.88890130540430168255199223667, 4.00691147074192089685643714493, 4.05567267908212512396076764298, 4.24389164228588813356651247667, 4.25310315379480467698331858439, 4.80528131891132812416345878566, 4.96809413726649574342654001295
Plot not available for L-functions of degree greater than 10.
