| L(s) = 1 | − 1.68·3-s + 5-s − 4.78·7-s − 0.157·9-s − 1.17·11-s + 3.55·13-s − 1.68·15-s + 5.24·17-s + 2.24·19-s + 8.07·21-s − 8.41·23-s + 25-s + 5.32·27-s − 8.71·29-s + 3.38·31-s + 1.98·33-s − 4.78·35-s + 9.53·37-s − 5.98·39-s − 2.73·41-s + 4.70·43-s − 0.157·45-s − 11.7·47-s + 15.9·49-s − 8.83·51-s − 4.56·53-s − 1.17·55-s + ⋯ |
| L(s) = 1 | − 0.973·3-s + 0.447·5-s − 1.81·7-s − 0.0526·9-s − 0.355·11-s + 0.984·13-s − 0.435·15-s + 1.27·17-s + 0.515·19-s + 1.76·21-s − 1.75·23-s + 0.200·25-s + 1.02·27-s − 1.61·29-s + 0.608·31-s + 0.346·33-s − 0.809·35-s + 1.56·37-s − 0.958·39-s − 0.427·41-s + 0.718·43-s − 0.0235·45-s − 1.70·47-s + 2.27·49-s − 1.23·51-s − 0.626·53-s − 0.158·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 151 | \( 1 - T \) |
| good | 3 | \( 1 + 1.68T + 3T^{2} \) |
| 7 | \( 1 + 4.78T + 7T^{2} \) |
| 11 | \( 1 + 1.17T + 11T^{2} \) |
| 13 | \( 1 - 3.55T + 13T^{2} \) |
| 17 | \( 1 - 5.24T + 17T^{2} \) |
| 19 | \( 1 - 2.24T + 19T^{2} \) |
| 23 | \( 1 + 8.41T + 23T^{2} \) |
| 29 | \( 1 + 8.71T + 29T^{2} \) |
| 31 | \( 1 - 3.38T + 31T^{2} \) |
| 37 | \( 1 - 9.53T + 37T^{2} \) |
| 41 | \( 1 + 2.73T + 41T^{2} \) |
| 43 | \( 1 - 4.70T + 43T^{2} \) |
| 47 | \( 1 + 11.7T + 47T^{2} \) |
| 53 | \( 1 + 4.56T + 53T^{2} \) |
| 59 | \( 1 - 2.95T + 59T^{2} \) |
| 61 | \( 1 - 3.98T + 61T^{2} \) |
| 67 | \( 1 - 13.8T + 67T^{2} \) |
| 71 | \( 1 + 3.61T + 71T^{2} \) |
| 73 | \( 1 - 9.48T + 73T^{2} \) |
| 79 | \( 1 - 1.13T + 79T^{2} \) |
| 83 | \( 1 - 15.4T + 83T^{2} \) |
| 89 | \( 1 + 0.962T + 89T^{2} \) |
| 97 | \( 1 - 3.59T + 97T^{2} \) |
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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
−7.70280953941152396887441647764, −6.66970960092849035350591756671, −6.16061641758384371620806799827, −5.80250923931541904504590130203, −5.15154693883881559372508898311, −3.85602601733083868207878617878, −3.35175661579419837652679891517, −2.39838588318668420463097765486, −1.02611236756747229705119350687, 0,
1.02611236756747229705119350687, 2.39838588318668420463097765486, 3.35175661579419837652679891517, 3.85602601733083868207878617878, 5.15154693883881559372508898311, 5.80250923931541904504590130203, 6.16061641758384371620806799827, 6.66970960092849035350591756671, 7.70280953941152396887441647764
