| L(s) = 1 | − 1.17·2-s − 0.618·4-s + 0.618·5-s + 0.618·7-s + 3.07·8-s − 0.726·10-s − 4.23·11-s − 0.449·13-s − 0.726·14-s − 2.38·16-s + 4.23·17-s − 0.381·20-s + 4.97·22-s + 3.76·23-s − 4.61·25-s + 0.527·26-s − 0.381·28-s − 5.25·29-s − 3.35·31-s − 3.35·32-s − 4.97·34-s + 0.381·35-s + 8.50·37-s + 1.90·40-s + 5.70·41-s − 3.76·43-s + 2.61·44-s + ⋯ |
| L(s) = 1 | − 0.831·2-s − 0.309·4-s + 0.276·5-s + 0.233·7-s + 1.08·8-s − 0.229·10-s − 1.27·11-s − 0.124·13-s − 0.194·14-s − 0.595·16-s + 1.02·17-s − 0.0854·20-s + 1.06·22-s + 0.784·23-s − 0.923·25-s + 0.103·26-s − 0.0721·28-s − 0.976·29-s − 0.602·31-s − 0.593·32-s − 0.854·34-s + 0.0645·35-s + 1.39·37-s + 0.300·40-s + 0.891·41-s − 0.573·43-s + 0.394·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 1.17T + 2T^{2} \) |
| 5 | \( 1 - 0.618T + 5T^{2} \) |
| 7 | \( 1 - 0.618T + 7T^{2} \) |
| 11 | \( 1 + 4.23T + 11T^{2} \) |
| 13 | \( 1 + 0.449T + 13T^{2} \) |
| 17 | \( 1 - 4.23T + 17T^{2} \) |
| 23 | \( 1 - 3.76T + 23T^{2} \) |
| 29 | \( 1 + 5.25T + 29T^{2} \) |
| 31 | \( 1 + 3.35T + 31T^{2} \) |
| 37 | \( 1 - 8.50T + 37T^{2} \) |
| 41 | \( 1 - 5.70T + 41T^{2} \) |
| 43 | \( 1 + 3.76T + 43T^{2} \) |
| 47 | \( 1 + 10.6T + 47T^{2} \) |
| 53 | \( 1 - 10.1T + 53T^{2} \) |
| 59 | \( 1 - 12.8T + 59T^{2} \) |
| 61 | \( 1 + 7.23T + 61T^{2} \) |
| 67 | \( 1 + 10.2T + 67T^{2} \) |
| 71 | \( 1 - 2.80T + 71T^{2} \) |
| 73 | \( 1 + 12.7T + 73T^{2} \) |
| 79 | \( 1 - 13.0T + 79T^{2} \) |
| 83 | \( 1 + 7.94T + 83T^{2} \) |
| 89 | \( 1 + 9.95T + 89T^{2} \) |
| 97 | \( 1 + 2.90T + 97T^{2} \) |
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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
−8.212796704563196820732279860238, −7.73076883548608585488868066338, −7.16456722232693337065932452832, −5.86661995163424643678092228342, −5.28428876171382602404451082025, −4.50455178983022977095793581063, −3.44619558742555555209945752592, −2.34027694510851290299675886127, −1.28392907614517624756559696529, 0,
1.28392907614517624756559696529, 2.34027694510851290299675886127, 3.44619558742555555209945752592, 4.50455178983022977095793581063, 5.28428876171382602404451082025, 5.86661995163424643678092228342, 7.16456722232693337065932452832, 7.73076883548608585488868066338, 8.212796704563196820732279860238
