L(s) = 1 | + 3-s + 5-s + 9-s + 11-s − 2·13-s + 15-s − 2·17-s − 4·19-s + 4·23-s + 25-s + 27-s + 2·29-s − 4·31-s + 33-s + 6·37-s − 2·39-s − 2·43-s + 45-s − 8·47-s − 2·51-s − 6·53-s + 55-s − 4·57-s − 2·61-s − 2·65-s + 8·67-s + 4·69-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1/3·9-s + 0.301·11-s − 0.554·13-s + 0.258·15-s − 0.485·17-s − 0.917·19-s + 0.834·23-s + 1/5·25-s + 0.192·27-s + 0.371·29-s − 0.718·31-s + 0.174·33-s + 0.986·37-s − 0.320·39-s − 0.304·43-s + 0.149·45-s − 1.16·47-s − 0.280·51-s − 0.824·53-s + 0.134·55-s − 0.529·57-s − 0.256·61-s − 0.248·65-s + 0.977·67-s + 0.481·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.732301407\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.732301407\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.40414455346735, −13.67951504338394, −13.21870546413349, −12.80021346597888, −12.42961023045289, −11.64455636566923, −11.15078775559628, −10.71193861109538, −9.971340225653993, −9.687694256720034, −9.082395314086741, −8.617049135300128, −8.163150363941462, −7.451247467713634, −6.927748984672218, −6.487281365041306, −5.838343621902525, −5.198266057383321, −4.452220676469230, −4.220699162507583, −3.155894873877371, −2.867128460464460, −1.992931762468407, −1.572231047602000, −0.5168104366276420,
0.5168104366276420, 1.572231047602000, 1.992931762468407, 2.867128460464460, 3.155894873877371, 4.220699162507583, 4.452220676469230, 5.198266057383321, 5.838343621902525, 6.487281365041306, 6.927748984672218, 7.451247467713634, 8.163150363941462, 8.617049135300128, 9.082395314086741, 9.687694256720034, 9.971340225653993, 10.71193861109538, 11.15078775559628, 11.64455636566923, 12.42961023045289, 12.80021346597888, 13.21870546413349, 13.67951504338394, 14.40414455346735