| L(s) = 1 | − 2.55·2-s − 2.76·3-s + 4.50·4-s + 0.117·5-s + 7.05·6-s − 0.649·7-s − 6.39·8-s + 4.64·9-s − 0.300·10-s − 3.37·11-s − 12.4·12-s − 4.83·13-s + 1.65·14-s − 0.325·15-s + 7.29·16-s − 2.19·17-s − 11.8·18-s − 2.67·19-s + 0.530·20-s + 1.79·21-s + 8.61·22-s + 6.36·23-s + 17.6·24-s − 4.98·25-s + 12.3·26-s − 4.54·27-s − 2.92·28-s + ⋯ |
| L(s) = 1 | − 1.80·2-s − 1.59·3-s + 2.25·4-s + 0.0526·5-s + 2.87·6-s − 0.245·7-s − 2.26·8-s + 1.54·9-s − 0.0949·10-s − 1.01·11-s − 3.59·12-s − 1.34·13-s + 0.442·14-s − 0.0840·15-s + 1.82·16-s − 0.531·17-s − 2.79·18-s − 0.614·19-s + 0.118·20-s + 0.391·21-s + 1.83·22-s + 1.32·23-s + 3.60·24-s − 0.997·25-s + 2.41·26-s − 0.874·27-s − 0.553·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.09195126679\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09195126679\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 \) |
| good | 2 | \( 1 + 2.55T + 2T^{2} \) |
| 3 | \( 1 + 2.76T + 3T^{2} \) |
| 5 | \( 1 - 0.117T + 5T^{2} \) |
| 7 | \( 1 + 0.649T + 7T^{2} \) |
| 11 | \( 1 + 3.37T + 11T^{2} \) |
| 13 | \( 1 + 4.83T + 13T^{2} \) |
| 17 | \( 1 + 2.19T + 17T^{2} \) |
| 19 | \( 1 + 2.67T + 19T^{2} \) |
| 23 | \( 1 - 6.36T + 23T^{2} \) |
| 29 | \( 1 - 0.967T + 29T^{2} \) |
| 31 | \( 1 + 1.96T + 31T^{2} \) |
| 37 | \( 1 + 9.69T + 37T^{2} \) |
| 41 | \( 1 - 5.20T + 41T^{2} \) |
| 47 | \( 1 + 4.11T + 47T^{2} \) |
| 53 | \( 1 + 8.39T + 53T^{2} \) |
| 59 | \( 1 - 10.8T + 59T^{2} \) |
| 61 | \( 1 - 0.117T + 61T^{2} \) |
| 67 | \( 1 + 11.7T + 67T^{2} \) |
| 71 | \( 1 - 0.127T + 71T^{2} \) |
| 73 | \( 1 + 11.8T + 73T^{2} \) |
| 79 | \( 1 + 3.46T + 79T^{2} \) |
| 83 | \( 1 + 13.2T + 83T^{2} \) |
| 89 | \( 1 - 6.58T + 89T^{2} \) |
| 97 | \( 1 - 7.76T + 97T^{2} \) |
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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
−9.413898992099467511656383498351, −8.550189558208779750090904843231, −7.55625714234833263896995298261, −7.05209000622343435828843412760, −6.33835329449771645435387077118, −5.44202463080490765898893275877, −4.65144216077232179554925354978, −2.82070077638495007364145717678, −1.73033362577728409722368450060, −0.28115813082282969021024612136,
0.28115813082282969021024612136, 1.73033362577728409722368450060, 2.82070077638495007364145717678, 4.65144216077232179554925354978, 5.44202463080490765898893275877, 6.33835329449771645435387077118, 7.05209000622343435828843412760, 7.55625714234833263896995298261, 8.550189558208779750090904843231, 9.413898992099467511656383498351
