| L(s) = 1 | + 3-s + 4.20·5-s − 0.497·7-s + 9-s − 0.310·11-s − 0.223·13-s + 4.20·15-s + 3.33·17-s − 0.261·19-s − 0.497·21-s − 9.01·23-s + 12.6·25-s + 27-s + 3.87·29-s + 0.329·31-s − 0.310·33-s − 2.09·35-s − 0.490·37-s − 0.223·39-s + 9.61·41-s + 2.34·43-s + 4.20·45-s + 9.94·47-s − 6.75·49-s + 3.33·51-s + 2.93·53-s − 1.30·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.87·5-s − 0.188·7-s + 0.333·9-s − 0.0936·11-s − 0.0619·13-s + 1.08·15-s + 0.808·17-s − 0.0598·19-s − 0.108·21-s − 1.88·23-s + 2.52·25-s + 0.192·27-s + 0.719·29-s + 0.0592·31-s − 0.0540·33-s − 0.353·35-s − 0.0806·37-s − 0.0357·39-s + 1.50·41-s + 0.357·43-s + 0.626·45-s + 1.45·47-s − 0.964·49-s + 0.466·51-s + 0.403·53-s − 0.175·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.544803389\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.544803389\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 4.20T + 5T^{2} \) |
| 7 | \( 1 + 0.497T + 7T^{2} \) |
| 11 | \( 1 + 0.310T + 11T^{2} \) |
| 13 | \( 1 + 0.223T + 13T^{2} \) |
| 17 | \( 1 - 3.33T + 17T^{2} \) |
| 19 | \( 1 + 0.261T + 19T^{2} \) |
| 23 | \( 1 + 9.01T + 23T^{2} \) |
| 29 | \( 1 - 3.87T + 29T^{2} \) |
| 31 | \( 1 - 0.329T + 31T^{2} \) |
| 37 | \( 1 + 0.490T + 37T^{2} \) |
| 41 | \( 1 - 9.61T + 41T^{2} \) |
| 43 | \( 1 - 2.34T + 43T^{2} \) |
| 47 | \( 1 - 9.94T + 47T^{2} \) |
| 53 | \( 1 - 2.93T + 53T^{2} \) |
| 59 | \( 1 - 2.69T + 59T^{2} \) |
| 61 | \( 1 + 1.23T + 61T^{2} \) |
| 67 | \( 1 - 11.6T + 67T^{2} \) |
| 71 | \( 1 + 3.61T + 71T^{2} \) |
| 73 | \( 1 + 4.33T + 73T^{2} \) |
| 79 | \( 1 + 2.59T + 79T^{2} \) |
| 83 | \( 1 - 3.06T + 83T^{2} \) |
| 89 | \( 1 - 14.4T + 89T^{2} \) |
| 97 | \( 1 + 0.603T + 97T^{2} \) |
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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
−8.533874554480922910834973151133, −7.79839862426895854200764360855, −6.91049637149287000275372962213, −6.05244659912240677164009051120, −5.72186471780727875939668972215, −4.75093051675081321243213710093, −3.75291152533358043720562357782, −2.66401591591115682683405602012, −2.14190000911652709267965327465, −1.12449216975346242392262283508,
1.12449216975346242392262283508, 2.14190000911652709267965327465, 2.66401591591115682683405602012, 3.75291152533358043720562357782, 4.75093051675081321243213710093, 5.72186471780727875939668972215, 6.05244659912240677164009051120, 6.91049637149287000275372962213, 7.79839862426895854200764360855, 8.533874554480922910834973151133
