| L(s) = 1 | − 0.656·2-s − 3-s − 1.56·4-s + 0.656·6-s + 1.34·7-s + 2.34·8-s + 9-s + 4.16·11-s + 1.56·12-s + 3.56·13-s − 0.882·14-s + 1.59·16-s + 0.744·17-s − 0.656·18-s − 1.74·19-s − 1.34·21-s − 2.73·22-s + 4.25·23-s − 2.34·24-s − 2.34·26-s − 27-s − 2.10·28-s + 1.68·29-s + 8.13·31-s − 5.73·32-s − 4.16·33-s − 0.488·34-s + ⋯ |
| L(s) = 1 | − 0.464·2-s − 0.577·3-s − 0.784·4-s + 0.268·6-s + 0.507·7-s + 0.828·8-s + 0.333·9-s + 1.25·11-s + 0.452·12-s + 0.989·13-s − 0.235·14-s + 0.399·16-s + 0.180·17-s − 0.154·18-s − 0.400·19-s − 0.293·21-s − 0.583·22-s + 0.887·23-s − 0.478·24-s − 0.459·26-s − 0.192·27-s − 0.398·28-s + 0.313·29-s + 1.46·31-s − 1.01·32-s − 0.725·33-s − 0.0838·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.218467157\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.218467157\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 2 | \( 1 + 0.656T + 2T^{2} \) |
| 7 | \( 1 - 1.34T + 7T^{2} \) |
| 11 | \( 1 - 4.16T + 11T^{2} \) |
| 13 | \( 1 - 3.56T + 13T^{2} \) |
| 17 | \( 1 - 0.744T + 17T^{2} \) |
| 19 | \( 1 + 1.74T + 19T^{2} \) |
| 23 | \( 1 - 4.25T + 23T^{2} \) |
| 29 | \( 1 - 1.68T + 29T^{2} \) |
| 31 | \( 1 - 8.13T + 31T^{2} \) |
| 41 | \( 1 + 11.6T + 41T^{2} \) |
| 43 | \( 1 - 8.39T + 43T^{2} \) |
| 47 | \( 1 - 8.79T + 47T^{2} \) |
| 53 | \( 1 - 1.48T + 53T^{2} \) |
| 59 | \( 1 + 11.0T + 59T^{2} \) |
| 61 | \( 1 + 5.59T + 61T^{2} \) |
| 67 | \( 1 + 4.28T + 67T^{2} \) |
| 71 | \( 1 + 1.70T + 71T^{2} \) |
| 73 | \( 1 + 4.87T + 73T^{2} \) |
| 79 | \( 1 + 7.96T + 79T^{2} \) |
| 83 | \( 1 - 7.56T + 83T^{2} \) |
| 89 | \( 1 - 5.37T + 89T^{2} \) |
| 97 | \( 1 + 1.13T + 97T^{2} \) |
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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
−8.832891021362214368684938544188, −8.255670991704864577590975822820, −7.36275371446768429100846405560, −6.48439673935586207690583876472, −5.80591167834084812310834661811, −4.74502174233656399797012915925, −4.28353568652989496936580838652, −3.29437087395378053639196467466, −1.58809590669056932816346671060, −0.853956373311930044073455985290,
0.853956373311930044073455985290, 1.58809590669056932816346671060, 3.29437087395378053639196467466, 4.28353568652989496936580838652, 4.74502174233656399797012915925, 5.80591167834084812310834661811, 6.48439673935586207690583876472, 7.36275371446768429100846405560, 8.255670991704864577590975822820, 8.832891021362214368684938544188
