| L(s) = 1 | − 3.41·3-s + 5-s − 2·7-s + 8.65·9-s + 4.24·11-s − 13-s − 3.41·15-s − 4.82·17-s − 8.24·19-s + 6.82·21-s + 5.07·23-s + 25-s − 19.3·27-s − 9.65·29-s − 1.41·31-s − 14.4·33-s − 2·35-s − 1.17·37-s + 3.41·39-s − 0.828·41-s − 1.75·43-s + 8.65·45-s + 2·47-s − 3·49-s + 16.4·51-s − 3.17·53-s + 4.24·55-s + ⋯ |
| L(s) = 1 | − 1.97·3-s + 0.447·5-s − 0.755·7-s + 2.88·9-s + 1.27·11-s − 0.277·13-s − 0.881·15-s − 1.17·17-s − 1.89·19-s + 1.49·21-s + 1.05·23-s + 0.200·25-s − 3.71·27-s − 1.79·29-s − 0.254·31-s − 2.52·33-s − 0.338·35-s − 0.192·37-s + 0.546·39-s − 0.129·41-s − 0.267·43-s + 1.29·45-s + 0.291·47-s − 0.428·49-s + 2.30·51-s − 0.435·53-s + 0.572·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + 3.41T + 3T^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 - 4.24T + 11T^{2} \) |
| 17 | \( 1 + 4.82T + 17T^{2} \) |
| 19 | \( 1 + 8.24T + 19T^{2} \) |
| 23 | \( 1 - 5.07T + 23T^{2} \) |
| 29 | \( 1 + 9.65T + 29T^{2} \) |
| 31 | \( 1 + 1.41T + 31T^{2} \) |
| 37 | \( 1 + 1.17T + 37T^{2} \) |
| 41 | \( 1 + 0.828T + 41T^{2} \) |
| 43 | \( 1 + 1.75T + 43T^{2} \) |
| 47 | \( 1 - 2T + 47T^{2} \) |
| 53 | \( 1 + 3.17T + 53T^{2} \) |
| 59 | \( 1 + 5.41T + 59T^{2} \) |
| 61 | \( 1 + 7.31T + 61T^{2} \) |
| 67 | \( 1 - 0.828T + 67T^{2} \) |
| 71 | \( 1 + 9.89T + 71T^{2} \) |
| 73 | \( 1 - 6.82T + 73T^{2} \) |
| 79 | \( 1 - 6.82T + 79T^{2} \) |
| 83 | \( 1 + 2T + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 - 0.343T + 97T^{2} \) |
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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
−10.76892515625415663970430565533, −9.678936856600928625873228805734, −9.012565739669430604925793247991, −7.14545383899240331564458120240, −6.51325824083471360453124345990, −6.00587786061643390440460691024, −4.83753720367894891817375253391, −3.96423568809408738269943956599, −1.74096157805781498233793892100, 0,
1.74096157805781498233793892100, 3.96423568809408738269943956599, 4.83753720367894891817375253391, 6.00587786061643390440460691024, 6.51325824083471360453124345990, 7.14545383899240331564458120240, 9.012565739669430604925793247991, 9.678936856600928625873228805734, 10.76892515625415663970430565533
