| L(s) = 1 | − 0.318·2-s − 1.89·4-s + 0.278·5-s + 1.24·8-s − 0.0887·10-s − 4.26·11-s − 13-s + 3.40·16-s + 6.27·17-s + 0.935·19-s − 0.529·20-s + 1.35·22-s + 0.554·23-s − 4.92·25-s + 0.318·26-s − 4.49·29-s + 0.861·31-s − 3.56·32-s − 2·34-s + 4.40·37-s − 0.297·38-s + 0.346·40-s − 0.979·41-s − 1.71·43-s + 8.10·44-s − 0.176·46-s + 4.42·47-s + ⋯ |
| L(s) = 1 | − 0.225·2-s − 0.949·4-s + 0.124·5-s + 0.439·8-s − 0.0280·10-s − 1.28·11-s − 0.277·13-s + 0.850·16-s + 1.52·17-s + 0.214·19-s − 0.118·20-s + 0.289·22-s + 0.115·23-s − 0.984·25-s + 0.0624·26-s − 0.834·29-s + 0.154·31-s − 0.630·32-s − 0.342·34-s + 0.723·37-s − 0.0483·38-s + 0.0547·40-s − 0.152·41-s − 0.261·43-s + 1.22·44-s − 0.0260·46-s + 0.645·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5733 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5733 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 + 0.318T + 2T^{2} \) |
| 5 | \( 1 - 0.278T + 5T^{2} \) |
| 11 | \( 1 + 4.26T + 11T^{2} \) |
| 17 | \( 1 - 6.27T + 17T^{2} \) |
| 19 | \( 1 - 0.935T + 19T^{2} \) |
| 23 | \( 1 - 0.554T + 23T^{2} \) |
| 29 | \( 1 + 4.49T + 29T^{2} \) |
| 31 | \( 1 - 0.861T + 31T^{2} \) |
| 37 | \( 1 - 4.40T + 37T^{2} \) |
| 41 | \( 1 + 0.979T + 41T^{2} \) |
| 43 | \( 1 + 1.71T + 43T^{2} \) |
| 47 | \( 1 - 4.42T + 47T^{2} \) |
| 53 | \( 1 - 9.30T + 53T^{2} \) |
| 59 | \( 1 + 12.5T + 59T^{2} \) |
| 61 | \( 1 - 1.97T + 61T^{2} \) |
| 67 | \( 1 + 3.97T + 67T^{2} \) |
| 71 | \( 1 + 2.43T + 71T^{2} \) |
| 73 | \( 1 - 4.83T + 73T^{2} \) |
| 79 | \( 1 - 12.5T + 79T^{2} \) |
| 83 | \( 1 - 11.1T + 83T^{2} \) |
| 89 | \( 1 - 12.1T + 89T^{2} \) |
| 97 | \( 1 + 12.9T + 97T^{2} \) |
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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
−7.72372222917730613914257437633, −7.49272619491455962548511083510, −6.16784890277378893272271562395, −5.41704675096448458910168219690, −5.06421493009754782103870830313, −4.05072769600161448211115694408, −3.32029714764343935195519437863, −2.34560305823690313294178514303, −1.13649851124935513998502809582, 0,
1.13649851124935513998502809582, 2.34560305823690313294178514303, 3.32029714764343935195519437863, 4.05072769600161448211115694408, 5.06421493009754782103870830313, 5.41704675096448458910168219690, 6.16784890277378893272271562395, 7.49272619491455962548511083510, 7.72372222917730613914257437633
