| L(s) = 1 | + 1.87·3-s − 1.18·7-s + 0.532·9-s − 2.18·11-s + 1.71·13-s + 0.120·17-s + 19-s − 2.22·21-s − 7.98·23-s − 4.63·27-s + 3.24·29-s − 8.41·31-s − 4.10·33-s − 3.33·37-s + 3.22·39-s − 8.98·41-s − 4.06·43-s + 1.71·47-s − 5.59·49-s + 0.226·51-s + 6.51·53-s + 1.87·57-s + 10.2·59-s + 6.53·61-s − 0.630·63-s + 2.18·67-s − 15.0·69-s + ⋯ |
| L(s) = 1 | + 1.08·3-s − 0.447·7-s + 0.177·9-s − 0.658·11-s + 0.476·13-s + 0.0292·17-s + 0.229·19-s − 0.485·21-s − 1.66·23-s − 0.892·27-s + 0.603·29-s − 1.51·31-s − 0.714·33-s − 0.547·37-s + 0.516·39-s − 1.40·41-s − 0.619·43-s + 0.250·47-s − 0.799·49-s + 0.0317·51-s + 0.895·53-s + 0.248·57-s + 1.32·59-s + 0.837·61-s − 0.0794·63-s + 0.266·67-s − 1.80·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 1.87T + 3T^{2} \) |
| 7 | \( 1 + 1.18T + 7T^{2} \) |
| 11 | \( 1 + 2.18T + 11T^{2} \) |
| 13 | \( 1 - 1.71T + 13T^{2} \) |
| 17 | \( 1 - 0.120T + 17T^{2} \) |
| 23 | \( 1 + 7.98T + 23T^{2} \) |
| 29 | \( 1 - 3.24T + 29T^{2} \) |
| 31 | \( 1 + 8.41T + 31T^{2} \) |
| 37 | \( 1 + 3.33T + 37T^{2} \) |
| 41 | \( 1 + 8.98T + 41T^{2} \) |
| 43 | \( 1 + 4.06T + 43T^{2} \) |
| 47 | \( 1 - 1.71T + 47T^{2} \) |
| 53 | \( 1 - 6.51T + 53T^{2} \) |
| 59 | \( 1 - 10.2T + 59T^{2} \) |
| 61 | \( 1 - 6.53T + 61T^{2} \) |
| 67 | \( 1 - 2.18T + 67T^{2} \) |
| 71 | \( 1 + 9.12T + 71T^{2} \) |
| 73 | \( 1 + 0.773T + 73T^{2} \) |
| 79 | \( 1 - 1.63T + 79T^{2} \) |
| 83 | \( 1 + 2.44T + 83T^{2} \) |
| 89 | \( 1 - 2.83T + 89T^{2} \) |
| 97 | \( 1 - 2.19T + 97T^{2} \) |
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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
−8.375785220370528409422845892935, −7.53192541433283629983734259510, −6.79490506050379358829243719492, −5.86000348235428131809255678835, −5.18711814676500643671076240438, −3.94763254075985040351879470846, −3.44220821863665790200337104837, −2.55472612199139275649697105099, −1.72958467717516518021661859720, 0,
1.72958467717516518021661859720, 2.55472612199139275649697105099, 3.44220821863665790200337104837, 3.94763254075985040351879470846, 5.18711814676500643671076240438, 5.86000348235428131809255678835, 6.79490506050379358829243719492, 7.53192541433283629983734259510, 8.375785220370528409422845892935
