L(s) = 1 | − 2.50·2-s + 0.477·3-s + 4.27·4-s + 0.361·5-s − 1.19·6-s − 0.370·7-s − 5.69·8-s − 2.77·9-s − 0.906·10-s + 1.13·11-s + 2.03·12-s − 2.60·13-s + 0.929·14-s + 0.172·15-s + 5.72·16-s + 17-s + 6.94·18-s + 0.661·19-s + 1.54·20-s − 0.177·21-s − 2.84·22-s + 1.16·23-s − 2.71·24-s − 4.86·25-s + 6.51·26-s − 2.75·27-s − 1.58·28-s + ⋯ |
L(s) = 1 | − 1.77·2-s + 0.275·3-s + 2.13·4-s + 0.161·5-s − 0.487·6-s − 0.140·7-s − 2.01·8-s − 0.924·9-s − 0.286·10-s + 0.342·11-s + 0.588·12-s − 0.721·13-s + 0.248·14-s + 0.0445·15-s + 1.43·16-s + 0.242·17-s + 1.63·18-s + 0.151·19-s + 0.345·20-s − 0.0386·21-s − 0.606·22-s + 0.243·23-s − 0.554·24-s − 0.973·25-s + 1.27·26-s − 0.530·27-s − 0.299·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2669 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2669 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 17 | \( 1 - T \) |
| 157 | \( 1 - T \) |
good | 2 | \( 1 + 2.50T + 2T^{2} \) |
| 3 | \( 1 - 0.477T + 3T^{2} \) |
| 5 | \( 1 - 0.361T + 5T^{2} \) |
| 7 | \( 1 + 0.370T + 7T^{2} \) |
| 11 | \( 1 - 1.13T + 11T^{2} \) |
| 13 | \( 1 + 2.60T + 13T^{2} \) |
| 19 | \( 1 - 0.661T + 19T^{2} \) |
| 23 | \( 1 - 1.16T + 23T^{2} \) |
| 29 | \( 1 - 3.91T + 29T^{2} \) |
| 31 | \( 1 - 5.58T + 31T^{2} \) |
| 37 | \( 1 - 8.04T + 37T^{2} \) |
| 41 | \( 1 + 2.75T + 41T^{2} \) |
| 43 | \( 1 - 7.04T + 43T^{2} \) |
| 47 | \( 1 + 4.43T + 47T^{2} \) |
| 53 | \( 1 - 5.07T + 53T^{2} \) |
| 59 | \( 1 - 0.886T + 59T^{2} \) |
| 61 | \( 1 + 10.8T + 61T^{2} \) |
| 67 | \( 1 + 0.493T + 67T^{2} \) |
| 71 | \( 1 + 5.84T + 71T^{2} \) |
| 73 | \( 1 + 9.98T + 73T^{2} \) |
| 79 | \( 1 + 12.8T + 79T^{2} \) |
| 83 | \( 1 + 2.54T + 83T^{2} \) |
| 89 | \( 1 + 14.9T + 89T^{2} \) |
| 97 | \( 1 + 5.55T + 97T^{2} \) |
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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
−8.546117741390008124831859817883, −7.913703880011173222985227729739, −7.30253856706626663615169604738, −6.38830465647909018586970306591, −5.76487235846184707527354691524, −4.48234761535977625808714096689, −3.05608413760937754664629196828, −2.43884853858550758219266558545, −1.28018916159790273629230295482, 0,
1.28018916159790273629230295482, 2.43884853858550758219266558545, 3.05608413760937754664629196828, 4.48234761535977625808714096689, 5.76487235846184707527354691524, 6.38830465647909018586970306591, 7.30253856706626663615169604738, 7.913703880011173222985227729739, 8.546117741390008124831859817883