| L(s) = 1 | + 3-s + 5-s + 2.56·7-s + 9-s − 6.56·11-s + 13-s + 15-s + 5.68·17-s − 5.12·19-s + 2.56·21-s − 7.68·23-s + 25-s + 27-s − 7.12·29-s − 8·31-s − 6.56·33-s + 2.56·35-s − 3.43·37-s + 39-s + 3.43·41-s − 9.12·43-s + 45-s + 6.24·47-s − 0.438·49-s + 5.68·51-s − 10.8·53-s − 6.56·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.968·7-s + 0.333·9-s − 1.97·11-s + 0.277·13-s + 0.258·15-s + 1.37·17-s − 1.17·19-s + 0.558·21-s − 1.60·23-s + 0.200·25-s + 0.192·27-s − 1.32·29-s − 1.43·31-s − 1.14·33-s + 0.432·35-s − 0.565·37-s + 0.160·39-s + 0.536·41-s − 1.39·43-s + 0.149·45-s + 0.911·47-s − 0.0626·49-s + 0.796·51-s − 1.48·53-s − 0.884·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - 2.56T + 7T^{2} \) |
| 11 | \( 1 + 6.56T + 11T^{2} \) |
| 17 | \( 1 - 5.68T + 17T^{2} \) |
| 19 | \( 1 + 5.12T + 19T^{2} \) |
| 23 | \( 1 + 7.68T + 23T^{2} \) |
| 29 | \( 1 + 7.12T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 3.43T + 37T^{2} \) |
| 41 | \( 1 - 3.43T + 41T^{2} \) |
| 43 | \( 1 + 9.12T + 43T^{2} \) |
| 47 | \( 1 - 6.24T + 47T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 - 2.24T + 59T^{2} \) |
| 61 | \( 1 + 0.561T + 61T^{2} \) |
| 67 | \( 1 + 14.2T + 67T^{2} \) |
| 71 | \( 1 - 11.6T + 71T^{2} \) |
| 73 | \( 1 + 8.24T + 73T^{2} \) |
| 79 | \( 1 - 3.68T + 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 - 10.8T + 89T^{2} \) |
| 97 | \( 1 + 2.31T + 97T^{2} \) |
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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
−7.924361543530100215286370325370, −7.29471335713089919147068281395, −6.10529746409550916311978646708, −5.49045682962790697419638842511, −4.93752262600133308022721566262, −3.96587065026384040944160029227, −3.15350069737746840164369875611, −2.13683565219453368005474543705, −1.70249550514622880985164625874, 0,
1.70249550514622880985164625874, 2.13683565219453368005474543705, 3.15350069737746840164369875611, 3.96587065026384040944160029227, 4.93752262600133308022721566262, 5.49045682962790697419638842511, 6.10529746409550916311978646708, 7.29471335713089919147068281395, 7.924361543530100215286370325370
