L(s) = 1 | + 2·3-s + 4·5-s − 4·7-s + 2·9-s − 2·11-s + 4·13-s + 8·15-s − 8·17-s − 8·21-s + 8·23-s + 8·25-s + 6·27-s + 12·29-s + 12·31-s − 4·33-s − 16·35-s − 16·37-s + 8·39-s + 2·41-s + 8·45-s + 8·49-s − 16·51-s − 8·55-s − 8·61-s − 8·63-s + 16·65-s − 12·67-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 1.78·5-s − 1.51·7-s + 2/3·9-s − 0.603·11-s + 1.10·13-s + 2.06·15-s − 1.94·17-s − 1.74·21-s + 1.66·23-s + 8/5·25-s + 1.15·27-s + 2.22·29-s + 2.15·31-s − 0.696·33-s − 2.70·35-s − 2.63·37-s + 1.28·39-s + 0.312·41-s + 1.19·45-s + 8/7·49-s − 2.24·51-s − 1.07·55-s − 1.02·61-s − 1.00·63-s + 1.98·65-s − 1.46·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 73984 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73984 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.528412932\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.528412932\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 2 | | \( 1 \) |
| 17 | $C_2$ | \( 1 + 8 T + p T^{2} \) |
good | 3 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 16 T + 128 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.19105255786011038558167918405, −11.91868619711859985841790567706, −10.74774087792658603704220680865, −10.59868994929398975958987719165, −10.24553242668043582014254825631, −9.775810753819368657132614787286, −9.087427786252797888298616889355, −8.792517323354613124438410155995, −8.749347945729649817641521309181, −7.974298217133216531579590112059, −6.80270641536937640432218869098, −6.79943720552042142052863243359, −6.34248474431770863462830107853, −5.77130438582258132472085153298, −4.87838251434214152292920474000, −4.46140206064187737780235433887, −3.27382311528131386270005398283, −2.86398214961836819694560104162, −2.47923241691311400915744040490, −1.35637709071033249774807294574,
1.35637709071033249774807294574, 2.47923241691311400915744040490, 2.86398214961836819694560104162, 3.27382311528131386270005398283, 4.46140206064187737780235433887, 4.87838251434214152292920474000, 5.77130438582258132472085153298, 6.34248474431770863462830107853, 6.79943720552042142052863243359, 6.80270641536937640432218869098, 7.974298217133216531579590112059, 8.749347945729649817641521309181, 8.792517323354613124438410155995, 9.087427786252797888298616889355, 9.775810753819368657132614787286, 10.24553242668043582014254825631, 10.59868994929398975958987719165, 10.74774087792658603704220680865, 11.91868619711859985841790567706, 12.19105255786011038558167918405