| L(s) = 1 | − 1.10·2-s − 0.783·4-s − 0.105·5-s + 3.07·8-s + 0.116·10-s + 3.33·11-s − 2.47·13-s − 1.81·16-s − 1.61·17-s + 7.68·19-s + 0.0826·20-s − 3.68·22-s − 1.89·23-s − 4.98·25-s + 2.73·26-s − 9.29·29-s − 9.26·31-s − 4.13·32-s + 1.77·34-s − 1.98·37-s − 8.47·38-s − 0.323·40-s + 7.48·41-s + 7.54·43-s − 2.61·44-s + 2.09·46-s + 3.19·47-s + ⋯ |
| L(s) = 1 | − 0.779·2-s − 0.391·4-s − 0.0471·5-s + 1.08·8-s + 0.0367·10-s + 1.00·11-s − 0.687·13-s − 0.454·16-s − 0.391·17-s + 1.76·19-s + 0.0184·20-s − 0.784·22-s − 0.395·23-s − 0.997·25-s + 0.536·26-s − 1.72·29-s − 1.66·31-s − 0.730·32-s + 0.305·34-s − 0.325·37-s − 1.37·38-s − 0.0511·40-s + 1.16·41-s + 1.15·43-s − 0.394·44-s + 0.308·46-s + 0.466·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 1.10T + 2T^{2} \) |
| 5 | \( 1 + 0.105T + 5T^{2} \) |
| 11 | \( 1 - 3.33T + 11T^{2} \) |
| 13 | \( 1 + 2.47T + 13T^{2} \) |
| 17 | \( 1 + 1.61T + 17T^{2} \) |
| 19 | \( 1 - 7.68T + 19T^{2} \) |
| 23 | \( 1 + 1.89T + 23T^{2} \) |
| 29 | \( 1 + 9.29T + 29T^{2} \) |
| 31 | \( 1 + 9.26T + 31T^{2} \) |
| 37 | \( 1 + 1.98T + 37T^{2} \) |
| 41 | \( 1 - 7.48T + 41T^{2} \) |
| 43 | \( 1 - 7.54T + 43T^{2} \) |
| 47 | \( 1 - 3.19T + 47T^{2} \) |
| 53 | \( 1 + 9.97T + 53T^{2} \) |
| 59 | \( 1 + 4.45T + 59T^{2} \) |
| 61 | \( 1 - 5.67T + 61T^{2} \) |
| 67 | \( 1 - 9.97T + 67T^{2} \) |
| 71 | \( 1 - 3.29T + 71T^{2} \) |
| 73 | \( 1 - 4.72T + 73T^{2} \) |
| 79 | \( 1 - 7.69T + 79T^{2} \) |
| 83 | \( 1 + 1.16T + 83T^{2} \) |
| 89 | \( 1 + 6.02T + 89T^{2} \) |
| 97 | \( 1 + 3.80T + 97T^{2} \) |
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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
−7.993689098872107855818343428315, −7.54631037894647963570640719868, −6.94130145372319564391334491290, −5.76240288203224440449411273616, −5.19100669414064231585775557240, −4.10797257679142616922564411965, −3.61844408760811400771679022598, −2.18006143330922281298907267907, −1.25129122661590616766574193921, 0,
1.25129122661590616766574193921, 2.18006143330922281298907267907, 3.61844408760811400771679022598, 4.10797257679142616922564411965, 5.19100669414064231585775557240, 5.76240288203224440449411273616, 6.94130145372319564391334491290, 7.54631037894647963570640719868, 7.993689098872107855818343428315
