| L(s) = 1 | − 2-s + 0.274·3-s + 4-s + 0.892·5-s − 0.274·6-s + 2.55·7-s − 8-s − 2.92·9-s − 0.892·10-s − 11-s + 0.274·12-s + 2.80·13-s − 2.55·14-s + 0.245·15-s + 16-s + 2.51·17-s + 2.92·18-s + 0.892·20-s + 0.701·21-s + 22-s − 4.65·23-s − 0.274·24-s − 4.20·25-s − 2.80·26-s − 1.62·27-s + 2.55·28-s + 4.95·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.158·3-s + 0.5·4-s + 0.399·5-s − 0.112·6-s + 0.965·7-s − 0.353·8-s − 0.974·9-s − 0.282·10-s − 0.301·11-s + 0.0792·12-s + 0.777·13-s − 0.682·14-s + 0.0632·15-s + 0.250·16-s + 0.610·17-s + 0.689·18-s + 0.199·20-s + 0.153·21-s + 0.213·22-s − 0.971·23-s − 0.0560·24-s − 0.840·25-s − 0.549·26-s − 0.313·27-s + 0.482·28-s + 0.920·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.704556415\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.704556415\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 0.274T + 3T^{2} \) |
| 5 | \( 1 - 0.892T + 5T^{2} \) |
| 7 | \( 1 - 2.55T + 7T^{2} \) |
| 13 | \( 1 - 2.80T + 13T^{2} \) |
| 17 | \( 1 - 2.51T + 17T^{2} \) |
| 23 | \( 1 + 4.65T + 23T^{2} \) |
| 29 | \( 1 - 4.95T + 29T^{2} \) |
| 31 | \( 1 + 6.85T + 31T^{2} \) |
| 37 | \( 1 - 6.39T + 37T^{2} \) |
| 41 | \( 1 - 11.0T + 41T^{2} \) |
| 43 | \( 1 + 0.288T + 43T^{2} \) |
| 47 | \( 1 + 4.66T + 47T^{2} \) |
| 53 | \( 1 - 12.4T + 53T^{2} \) |
| 59 | \( 1 + 6.23T + 59T^{2} \) |
| 61 | \( 1 - 3.39T + 61T^{2} \) |
| 67 | \( 1 + 7.74T + 67T^{2} \) |
| 71 | \( 1 + 2.71T + 71T^{2} \) |
| 73 | \( 1 - 6.41T + 73T^{2} \) |
| 79 | \( 1 - 14.5T + 79T^{2} \) |
| 83 | \( 1 + 3.05T + 83T^{2} \) |
| 89 | \( 1 + 0.997T + 89T^{2} \) |
| 97 | \( 1 - 13.2T + 97T^{2} \) |
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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
−7.85435023746487202941053067009, −7.55524134010647696116242220664, −6.31991082436999608747053908151, −5.87671886265719335900203345219, −5.26155774886352268758008008217, −4.24228278170240703172640747392, −3.36858760485336244568314503149, −2.43073277151161422841424371625, −1.77567473624622087683125183683, −0.71927805363189081686457828772,
0.71927805363189081686457828772, 1.77567473624622087683125183683, 2.43073277151161422841424371625, 3.36858760485336244568314503149, 4.24228278170240703172640747392, 5.26155774886352268758008008217, 5.87671886265719335900203345219, 6.31991082436999608747053908151, 7.55524134010647696116242220664, 7.85435023746487202941053067009
