| L(s) = 1 | + 2-s − 3-s + 4-s − 2.85·5-s − 6-s + 1.31·7-s + 8-s + 9-s − 2.85·10-s + 4.30·11-s − 12-s + 2.80·13-s + 1.31·14-s + 2.85·15-s + 16-s + 2·17-s + 18-s − 19-s − 2.85·20-s − 1.31·21-s + 4.30·22-s + 5.16·23-s − 24-s + 3.15·25-s + 2.80·26-s − 27-s + 1.31·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.27·5-s − 0.408·6-s + 0.497·7-s + 0.353·8-s + 0.333·9-s − 0.903·10-s + 1.29·11-s − 0.288·12-s + 0.778·13-s + 0.351·14-s + 0.737·15-s + 0.250·16-s + 0.485·17-s + 0.235·18-s − 0.229·19-s − 0.638·20-s − 0.287·21-s + 0.918·22-s + 1.07·23-s − 0.204·24-s + 0.631·25-s + 0.550·26-s − 0.192·27-s + 0.248·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.539532879\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.539532879\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 53 | \( 1 + T \) |
| good | 5 | \( 1 + 2.85T + 5T^{2} \) |
| 7 | \( 1 - 1.31T + 7T^{2} \) |
| 11 | \( 1 - 4.30T + 11T^{2} \) |
| 13 | \( 1 - 2.80T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 23 | \( 1 - 5.16T + 23T^{2} \) |
| 29 | \( 1 + 3.16T + 29T^{2} \) |
| 31 | \( 1 - 9.53T + 31T^{2} \) |
| 37 | \( 1 + 8.51T + 37T^{2} \) |
| 41 | \( 1 - 2.70T + 41T^{2} \) |
| 43 | \( 1 - 1.32T + 43T^{2} \) |
| 47 | \( 1 + 6.86T + 47T^{2} \) |
| 59 | \( 1 - 6.09T + 59T^{2} \) |
| 61 | \( 1 + 8.09T + 61T^{2} \) |
| 67 | \( 1 + 10.6T + 67T^{2} \) |
| 71 | \( 1 + 0.502T + 71T^{2} \) |
| 73 | \( 1 - 0.925T + 73T^{2} \) |
| 79 | \( 1 - 6.22T + 79T^{2} \) |
| 83 | \( 1 - 14.1T + 83T^{2} \) |
| 89 | \( 1 - 12.0T + 89T^{2} \) |
| 97 | \( 1 - 7.22T + 97T^{2} \) |
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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
−7.947265593738216607693933515900, −7.24885846241925324791134974719, −6.56456656161002591648128854293, −6.01222211618912700364331586765, −4.96794235746575225855415373036, −4.49869857995269793307717339219, −3.71106988000239680676672210684, −3.21704708735095427292541432959, −1.70935663059450744300626561836, −0.826214132017443023990414305613,
0.826214132017443023990414305613, 1.70935663059450744300626561836, 3.21704708735095427292541432959, 3.71106988000239680676672210684, 4.49869857995269793307717339219, 4.96794235746575225855415373036, 6.01222211618912700364331586765, 6.56456656161002591648128854293, 7.24885846241925324791134974719, 7.947265593738216607693933515900
