| L(s) = 1 | − 4·9-s + 72·29-s − 24·41-s + 24·49-s − 26·81-s − 96·101-s − 88·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯ |
| L(s) = 1 | − 4/3·9-s + 13.3·29-s − 3.74·41-s + 24/7·49-s − 2.88·81-s − 9.55·101-s − 8·121-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 + 4/13·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{8} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{8} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.422562884\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.422562884\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 23 | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| good | 3 | \( ( 1 + T^{2} + p^{2} T^{4} )^{4} \) |
| 7 | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{4} \) |
| 11 | \( ( 1 + p T^{2} )^{8} \) |
| 13 | \( ( 1 - T^{2} + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 + 24 T^{2} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 - 12 T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 - 9 T + p T^{2} )^{8} \) |
| 31 | \( ( 1 - 17 T^{2} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 + p T^{2} )^{8} \) |
| 41 | \( ( 1 + 3 T + p T^{2} )^{8} \) |
| 43 | \( ( 1 - 84 T^{2} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 + 49 T^{2} + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 + 66 T^{2} + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 - 98 T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 - 32 T^{2} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 - 126 T^{2} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 - 97 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 + 79 T^{2} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 + 108 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{4} \) |
| 89 | \( ( 1 - 168 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 - 166 T^{2} + p^{2} T^{4} )^{4} \) |
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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
−4.09921021996700728511168770502, −3.86883632478825642349661267485, −3.47650163852783045054048012208, −3.44075438658072260046531515523, −3.28442520929020279282104616933, −3.21834619846016720560897581680, −3.13293391683494459282705670544, −2.87244567081197519444653261712, −2.84674273034664561521638067673, −2.64141457543551992901260100630, −2.57003286356525309209527726419, −2.53521732429237426541804023422, −2.52938244745078273887913216445, −2.47545840151407394914731372555, −2.40339157054862129876239266136, −1.82285683627161372738462519305, −1.55067138006863016524635276581, −1.37801035650357699717703156088, −1.36188594339038468361632552062, −1.18070876617604462435200750968, −1.17682978093125267313057398423, −1.01797766238544608743431260210, −0.63577927884418886561814381951, −0.45456748521317308201200426781, −0.22395134546757526020397620197,
0.22395134546757526020397620197, 0.45456748521317308201200426781, 0.63577927884418886561814381951, 1.01797766238544608743431260210, 1.17682978093125267313057398423, 1.18070876617604462435200750968, 1.36188594339038468361632552062, 1.37801035650357699717703156088, 1.55067138006863016524635276581, 1.82285683627161372738462519305, 2.40339157054862129876239266136, 2.47545840151407394914731372555, 2.52938244745078273887913216445, 2.53521732429237426541804023422, 2.57003286356525309209527726419, 2.64141457543551992901260100630, 2.84674273034664561521638067673, 2.87244567081197519444653261712, 3.13293391683494459282705670544, 3.21834619846016720560897581680, 3.28442520929020279282104616933, 3.44075438658072260046531515523, 3.47650163852783045054048012208, 3.86883632478825642349661267485, 4.09921021996700728511168770502
Plot not available for L-functions of degree greater than 10.
