| L(s) = 1 | + 3-s − 5-s + 9-s + 4·11-s + 2·13-s − 15-s − 2·17-s + 4·19-s + 8·23-s + 25-s + 27-s − 2·29-s + 4·33-s + 6·37-s + 2·39-s + 6·41-s + 4·43-s − 45-s − 2·51-s − 10·53-s − 4·55-s + 4·57-s + 12·59-s − 14·61-s − 2·65-s + 12·67-s + 8·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s + 1.20·11-s + 0.554·13-s − 0.258·15-s − 0.485·17-s + 0.917·19-s + 1.66·23-s + 1/5·25-s + 0.192·27-s − 0.371·29-s + 0.696·33-s + 0.986·37-s + 0.320·39-s + 0.937·41-s + 0.609·43-s − 0.149·45-s − 0.280·51-s − 1.37·53-s − 0.539·55-s + 0.529·57-s + 1.56·59-s − 1.79·61-s − 0.248·65-s + 1.46·67-s + 0.963·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.102129835\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.102129835\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−16.26158625921562, −15.84543601539716, −15.20294581811562, −14.70849924015988, −14.18203728292491, −13.67135354623125, −12.89587441223978, −12.59721548377167, −11.68881647798720, −11.19813910461926, −10.87317498589497, −9.677902214792115, −9.466387637789692, −8.735618317591577, −8.313139190968788, −7.371951328715601, −7.070494082123913, −6.274411071276473, −5.554190358748549, −4.575819234590486, −4.097742224731950, −3.307662455264993, −2.731199437603262, −1.555455497411807, −0.8532442649986090,
0.8532442649986090, 1.555455497411807, 2.731199437603262, 3.307662455264993, 4.097742224731950, 4.575819234590486, 5.554190358748549, 6.274411071276473, 7.070494082123913, 7.371951328715601, 8.313139190968788, 8.735618317591577, 9.466387637789692, 9.677902214792115, 10.87317498589497, 11.19813910461926, 11.68881647798720, 12.59721548377167, 12.89587441223978, 13.67135354623125, 14.18203728292491, 14.70849924015988, 15.20294581811562, 15.84543601539716, 16.26158625921562
