| L(s) = 1 | + 2-s + 1.90·3-s + 4-s − 3.81·5-s + 1.90·6-s + 1.69·7-s + 8-s + 0.616·9-s − 3.81·10-s + 3.78·11-s + 1.90·12-s + 0.0820·13-s + 1.69·14-s − 7.24·15-s + 16-s − 5.51·17-s + 0.616·18-s − 5.52·19-s − 3.81·20-s + 3.23·21-s + 3.78·22-s − 23-s + 1.90·24-s + 9.53·25-s + 0.0820·26-s − 4.53·27-s + 1.69·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.09·3-s + 0.5·4-s − 1.70·5-s + 0.776·6-s + 0.642·7-s + 0.353·8-s + 0.205·9-s − 1.20·10-s + 1.14·11-s + 0.548·12-s + 0.0227·13-s + 0.454·14-s − 1.87·15-s + 0.250·16-s − 1.33·17-s + 0.145·18-s − 1.26·19-s − 0.852·20-s + 0.705·21-s + 0.807·22-s − 0.208·23-s + 0.388·24-s + 1.90·25-s + 0.0160·26-s − 0.872·27-s + 0.321·28-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.90T + 3T^{2} \) |
| 5 | \( 1 + 3.81T + 5T^{2} \) |
| 7 | \( 1 - 1.69T + 7T^{2} \) |
| 11 | \( 1 - 3.78T + 11T^{2} \) |
| 13 | \( 1 - 0.0820T + 13T^{2} \) |
| 17 | \( 1 + 5.51T + 17T^{2} \) |
| 19 | \( 1 + 5.52T + 19T^{2} \) |
| 29 | \( 1 + 7.55T + 29T^{2} \) |
| 31 | \( 1 + 4.11T + 31T^{2} \) |
| 37 | \( 1 + 7.13T + 37T^{2} \) |
| 41 | \( 1 + 0.552T + 41T^{2} \) |
| 43 | \( 1 + 2.03T + 43T^{2} \) |
| 47 | \( 1 - 10.9T + 47T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 - 7.85T + 59T^{2} \) |
| 61 | \( 1 - 8.75T + 61T^{2} \) |
| 67 | \( 1 - 8.48T + 67T^{2} \) |
| 71 | \( 1 - 4.08T + 71T^{2} \) |
| 73 | \( 1 + 0.0747T + 73T^{2} \) |
| 79 | \( 1 + 13.0T + 79T^{2} \) |
| 83 | \( 1 - 8.10T + 83T^{2} \) |
| 89 | \( 1 - 8.42T + 89T^{2} \) |
| 97 | \( 1 + 8.84T + 97T^{2} \) |
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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
−7.77586063800275997380143254729, −7.06560271970789793566608288080, −6.54584512237509756397487142414, −5.37657183883914396352151848581, −4.43736699525293853739873082569, −3.87962819160913482584237436202, −3.61920052774297727574873837983, −2.46239320235718081878061523020, −1.69824414537848596759899663470, 0,
1.69824414537848596759899663470, 2.46239320235718081878061523020, 3.61920052774297727574873837983, 3.87962819160913482584237436202, 4.43736699525293853739873082569, 5.37657183883914396352151848581, 6.54584512237509756397487142414, 7.06560271970789793566608288080, 7.77586063800275997380143254729
