| L(s) = 1 | − 2-s + 1.72·3-s + 4-s − 1.24·5-s − 1.72·6-s − 3.18·7-s − 8-s − 0.0355·9-s + 1.24·10-s − 6.34·11-s + 1.72·12-s + 1.28·13-s + 3.18·14-s − 2.13·15-s + 16-s − 7.61·17-s + 0.0355·18-s − 1.60·19-s − 1.24·20-s − 5.48·21-s + 6.34·22-s + 23-s − 1.72·24-s − 3.46·25-s − 1.28·26-s − 5.22·27-s − 3.18·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.994·3-s + 0.5·4-s − 0.554·5-s − 0.702·6-s − 1.20·7-s − 0.353·8-s − 0.0118·9-s + 0.392·10-s − 1.91·11-s + 0.497·12-s + 0.356·13-s + 0.851·14-s − 0.551·15-s + 0.250·16-s − 1.84·17-s + 0.00837·18-s − 0.368·19-s − 0.277·20-s − 1.19·21-s + 1.35·22-s + 0.208·23-s − 0.351·24-s − 0.692·25-s − 0.252·26-s − 1.00·27-s − 0.602·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5140244471\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5140244471\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 131 | \( 1 + T \) |
| good | 3 | \( 1 - 1.72T + 3T^{2} \) |
| 5 | \( 1 + 1.24T + 5T^{2} \) |
| 7 | \( 1 + 3.18T + 7T^{2} \) |
| 11 | \( 1 + 6.34T + 11T^{2} \) |
| 13 | \( 1 - 1.28T + 13T^{2} \) |
| 17 | \( 1 + 7.61T + 17T^{2} \) |
| 19 | \( 1 + 1.60T + 19T^{2} \) |
| 29 | \( 1 - 7.55T + 29T^{2} \) |
| 31 | \( 1 - 3.43T + 31T^{2} \) |
| 37 | \( 1 - 11.2T + 37T^{2} \) |
| 41 | \( 1 + 10.1T + 41T^{2} \) |
| 43 | \( 1 - 2.40T + 43T^{2} \) |
| 47 | \( 1 + 11.5T + 47T^{2} \) |
| 53 | \( 1 + 9.78T + 53T^{2} \) |
| 59 | \( 1 - 13.8T + 59T^{2} \) |
| 61 | \( 1 - 4.31T + 61T^{2} \) |
| 67 | \( 1 + 5.06T + 67T^{2} \) |
| 71 | \( 1 - 9.71T + 71T^{2} \) |
| 73 | \( 1 - 0.706T + 73T^{2} \) |
| 79 | \( 1 - 9.76T + 79T^{2} \) |
| 83 | \( 1 + 4.79T + 83T^{2} \) |
| 89 | \( 1 - 11.3T + 89T^{2} \) |
| 97 | \( 1 - 3.40T + 97T^{2} \) |
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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
−8.226280511646740651850057082354, −7.71457122376980831572274637743, −6.66797458670737270049777141163, −6.34830952604211493538575128128, −5.19447180904094704000564102004, −4.26193908464365217325694942314, −3.29360207460368686533612237245, −2.72209535534140753426538742079, −2.15559349690262049282803510994, −0.36284558048016954501281753230,
0.36284558048016954501281753230, 2.15559349690262049282803510994, 2.72209535534140753426538742079, 3.29360207460368686533612237245, 4.26193908464365217325694942314, 5.19447180904094704000564102004, 6.34830952604211493538575128128, 6.66797458670737270049777141163, 7.71457122376980831572274637743, 8.226280511646740651850057082354
