| L(s) = 1 | + 2·3-s + 4·5-s − 6·7-s + 2·9-s + 4·11-s + 6·13-s + 8·15-s + 2·17-s − 12·21-s − 2·23-s + 11·25-s + 6·27-s + 8·33-s − 24·35-s − 2·37-s + 12·39-s + 20·41-s − 10·43-s + 8·45-s + 6·47-s + 18·49-s + 4·51-s + 10·53-s + 16·55-s − 12·63-s + 24·65-s + 2·67-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1.78·5-s − 2.26·7-s + 2/3·9-s + 1.20·11-s + 1.66·13-s + 2.06·15-s + 0.485·17-s − 2.61·21-s − 0.417·23-s + 11/5·25-s + 1.15·27-s + 1.39·33-s − 4.05·35-s − 0.328·37-s + 1.92·39-s + 3.12·41-s − 1.52·43-s + 1.19·45-s + 0.875·47-s + 18/7·49-s + 0.560·51-s + 1.37·53-s + 2.15·55-s − 1.51·63-s + 2.97·65-s + 0.244·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.973885177\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.973885177\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 138 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 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
−9.645350837740558908925893226743, −9.402456803584422082986961650440, −9.183588706578470632279672495318, −8.875916381237804046764419913818, −8.442967585645943387558307781780, −8.054915792416821113508114592346, −7.09784449636194623806615124941, −7.04691145765800249607407565859, −6.41447180530827700891439005324, −6.26544967810012028475264976236, −5.71220766864053706978570655988, −5.70035539540159393762998946890, −4.65789450072451515343506568620, −4.00085779896879103884871738354, −3.64492132258854567330998707155, −3.25428908236209704701738066116, −2.65187037517861713573738993739, −2.38533361247367005137780260175, −1.42990253219613563328555726280, −0.976448526410779716129716993104,
0.976448526410779716129716993104, 1.42990253219613563328555726280, 2.38533361247367005137780260175, 2.65187037517861713573738993739, 3.25428908236209704701738066116, 3.64492132258854567330998707155, 4.00085779896879103884871738354, 4.65789450072451515343506568620, 5.70035539540159393762998946890, 5.71220766864053706978570655988, 6.26544967810012028475264976236, 6.41447180530827700891439005324, 7.04691145765800249607407565859, 7.09784449636194623806615124941, 8.054915792416821113508114592346, 8.442967585645943387558307781780, 8.875916381237804046764419913818, 9.183588706578470632279672495318, 9.402456803584422082986961650440, 9.645350837740558908925893226743
