| L(s) = 1 | + 2.66·2-s + 5.08·4-s − 3.46·7-s + 8.21·8-s − 3.43·11-s − 5.70·13-s − 9.23·14-s + 11.6·16-s − 1.89·17-s + 2.46·19-s − 9.13·22-s − 8.84·23-s − 15.1·26-s − 17.6·28-s + 8.98·29-s − 1.90·31-s + 14.7·32-s − 5.03·34-s − 5.09·37-s + 6.57·38-s + 6.35·41-s + 3.43·43-s − 17.4·44-s − 23.5·46-s − 8.61·47-s + 5.02·49-s − 29.0·52-s + ⋯ |
| L(s) = 1 | + 1.88·2-s + 2.54·4-s − 1.31·7-s + 2.90·8-s − 1.03·11-s − 1.58·13-s − 2.46·14-s + 2.92·16-s − 0.458·17-s + 0.566·19-s − 1.94·22-s − 1.84·23-s − 2.97·26-s − 3.33·28-s + 1.66·29-s − 0.342·31-s + 2.60·32-s − 0.863·34-s − 0.837·37-s + 1.06·38-s + 0.991·41-s + 0.523·43-s − 2.63·44-s − 3.47·46-s − 1.25·47-s + 0.718·49-s − 4.02·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 2.66T + 2T^{2} \) |
| 7 | \( 1 + 3.46T + 7T^{2} \) |
| 11 | \( 1 + 3.43T + 11T^{2} \) |
| 13 | \( 1 + 5.70T + 13T^{2} \) |
| 17 | \( 1 + 1.89T + 17T^{2} \) |
| 19 | \( 1 - 2.46T + 19T^{2} \) |
| 23 | \( 1 + 8.84T + 23T^{2} \) |
| 29 | \( 1 - 8.98T + 29T^{2} \) |
| 31 | \( 1 + 1.90T + 31T^{2} \) |
| 37 | \( 1 + 5.09T + 37T^{2} \) |
| 41 | \( 1 - 6.35T + 41T^{2} \) |
| 43 | \( 1 - 3.43T + 43T^{2} \) |
| 47 | \( 1 + 8.61T + 47T^{2} \) |
| 53 | \( 1 + 2.03T + 53T^{2} \) |
| 59 | \( 1 + 8.47T + 59T^{2} \) |
| 61 | \( 1 + 7.90T + 61T^{2} \) |
| 67 | \( 1 + 6.95T + 67T^{2} \) |
| 71 | \( 1 - 2.91T + 71T^{2} \) |
| 73 | \( 1 - 11.9T + 73T^{2} \) |
| 79 | \( 1 + 6.60T + 79T^{2} \) |
| 83 | \( 1 + 12.9T + 83T^{2} \) |
| 89 | \( 1 + 11.5T + 89T^{2} \) |
| 97 | \( 1 - 2.17T + 97T^{2} \) |
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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
−7.45916122189663573977763917829, −6.80695132780004519846811084512, −6.16753003653636562686321798008, −5.55912553060422659057150113419, −4.78462592697742299613102452473, −4.23728694118630116445249246678, −3.18057824830046074389810054013, −2.78519159088471294911580580861, −1.99136196643060706463903585600, 0,
1.99136196643060706463903585600, 2.78519159088471294911580580861, 3.18057824830046074389810054013, 4.23728694118630116445249246678, 4.78462592697742299613102452473, 5.55912553060422659057150113419, 6.16753003653636562686321798008, 6.80695132780004519846811084512, 7.45916122189663573977763917829
