| L(s) = 1 | + 3-s − 2.76·5-s − 3.28·7-s + 9-s + 3.56·11-s − 4.67·13-s − 2.76·15-s + 6.82·17-s − 3.92·19-s − 3.28·21-s + 6.73·23-s + 2.62·25-s + 27-s + 6.59·29-s − 7.05·31-s + 3.56·33-s + 9.06·35-s − 3.23·37-s − 4.67·39-s − 6.47·41-s + 6.00·43-s − 2.76·45-s − 2.08·47-s + 3.79·49-s + 6.82·51-s + 2.26·53-s − 9.85·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.23·5-s − 1.24·7-s + 0.333·9-s + 1.07·11-s − 1.29·13-s − 0.712·15-s + 1.65·17-s − 0.901·19-s − 0.716·21-s + 1.40·23-s + 0.524·25-s + 0.192·27-s + 1.22·29-s − 1.26·31-s + 0.621·33-s + 1.53·35-s − 0.532·37-s − 0.749·39-s − 1.01·41-s + 0.915·43-s − 0.411·45-s − 0.303·47-s + 0.541·49-s + 0.956·51-s + 0.310·53-s − 1.32·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 + 2.76T + 5T^{2} \) |
| 7 | \( 1 + 3.28T + 7T^{2} \) |
| 11 | \( 1 - 3.56T + 11T^{2} \) |
| 13 | \( 1 + 4.67T + 13T^{2} \) |
| 17 | \( 1 - 6.82T + 17T^{2} \) |
| 19 | \( 1 + 3.92T + 19T^{2} \) |
| 23 | \( 1 - 6.73T + 23T^{2} \) |
| 29 | \( 1 - 6.59T + 29T^{2} \) |
| 31 | \( 1 + 7.05T + 31T^{2} \) |
| 37 | \( 1 + 3.23T + 37T^{2} \) |
| 41 | \( 1 + 6.47T + 41T^{2} \) |
| 43 | \( 1 - 6.00T + 43T^{2} \) |
| 47 | \( 1 + 2.08T + 47T^{2} \) |
| 53 | \( 1 - 2.26T + 53T^{2} \) |
| 59 | \( 1 - 2.05T + 59T^{2} \) |
| 61 | \( 1 - 4.28T + 61T^{2} \) |
| 67 | \( 1 + 7.96T + 67T^{2} \) |
| 71 | \( 1 - 3.87T + 71T^{2} \) |
| 73 | \( 1 - 4.15T + 73T^{2} \) |
| 79 | \( 1 + 1.39T + 79T^{2} \) |
| 83 | \( 1 - 4.70T + 83T^{2} \) |
| 89 | \( 1 - 3.98T + 89T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
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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
−7.35501395533333048938820422635, −7.02859329047515377892334661090, −6.33478145707466626712397360907, −5.28922069651709850684665019855, −4.49578260564909580185272370176, −3.63113995115555417687167490096, −3.34560601238500339138721461362, −2.47375447897725108939952483105, −1.12892515721894711854723166621, 0,
1.12892515721894711854723166621, 2.47375447897725108939952483105, 3.34560601238500339138721461362, 3.63113995115555417687167490096, 4.49578260564909580185272370176, 5.28922069651709850684665019855, 6.33478145707466626712397360907, 7.02859329047515377892334661090, 7.35501395533333048938820422635
