| L(s) = 1 | − 0.112·2-s − 3-s − 1.98·4-s + 2.19·5-s + 0.112·6-s + 0.450·8-s + 9-s − 0.247·10-s − 0.923·11-s + 1.98·12-s − 2.51·13-s − 2.19·15-s + 3.92·16-s − 5.19·17-s − 0.112·18-s + 4.74·19-s − 4.35·20-s + 0.104·22-s + 4.83·23-s − 0.450·24-s − 0.202·25-s + 0.283·26-s − 27-s + 5.73·29-s + 0.247·30-s − 8.74·31-s − 1.34·32-s + ⋯ |
| L(s) = 1 | − 0.0798·2-s − 0.577·3-s − 0.993·4-s + 0.979·5-s + 0.0461·6-s + 0.159·8-s + 0.333·9-s − 0.0782·10-s − 0.278·11-s + 0.573·12-s − 0.697·13-s − 0.565·15-s + 0.980·16-s − 1.25·17-s − 0.0266·18-s + 1.08·19-s − 0.973·20-s + 0.0222·22-s + 1.00·23-s − 0.0919·24-s − 0.0405·25-s + 0.0556·26-s − 0.192·27-s + 1.06·29-s + 0.0451·30-s − 1.56·31-s − 0.237·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 + 0.112T + 2T^{2} \) |
| 5 | \( 1 - 2.19T + 5T^{2} \) |
| 11 | \( 1 + 0.923T + 11T^{2} \) |
| 13 | \( 1 + 2.51T + 13T^{2} \) |
| 17 | \( 1 + 5.19T + 17T^{2} \) |
| 19 | \( 1 - 4.74T + 19T^{2} \) |
| 23 | \( 1 - 4.83T + 23T^{2} \) |
| 29 | \( 1 - 5.73T + 29T^{2} \) |
| 31 | \( 1 + 8.74T + 31T^{2} \) |
| 37 | \( 1 + 4.39T + 37T^{2} \) |
| 43 | \( 1 - 9.16T + 43T^{2} \) |
| 47 | \( 1 + 6.16T + 47T^{2} \) |
| 53 | \( 1 - 3.46T + 53T^{2} \) |
| 59 | \( 1 + 2.96T + 59T^{2} \) |
| 61 | \( 1 - 1.11T + 61T^{2} \) |
| 67 | \( 1 - 6.66T + 67T^{2} \) |
| 71 | \( 1 + 5.41T + 71T^{2} \) |
| 73 | \( 1 + 0.586T + 73T^{2} \) |
| 79 | \( 1 + 12.3T + 79T^{2} \) |
| 83 | \( 1 + 16.2T + 83T^{2} \) |
| 89 | \( 1 + 1.30T + 89T^{2} \) |
| 97 | \( 1 - 8.08T + 97T^{2} \) |
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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
−7.58000598233402189183912784479, −7.07228071283177283270562228238, −6.13584101678077199226519983867, −5.45557736507475775369356395527, −4.97405780222789780379497890907, −4.29960162201455916281792297786, −3.23079740928152214343934846091, −2.22334214636461401085301451645, −1.17487031644348674910864798787, 0,
1.17487031644348674910864798787, 2.22334214636461401085301451645, 3.23079740928152214343934846091, 4.29960162201455916281792297786, 4.97405780222789780379497890907, 5.45557736507475775369356395527, 6.13584101678077199226519983867, 7.07228071283177283270562228238, 7.58000598233402189183912784479
