L(s) = 1 | + 2.69·2-s − 1.59·3-s + 5.28·4-s − 1.39·5-s − 4.31·6-s − 4.07·7-s + 8.87·8-s − 0.450·9-s − 3.76·10-s + 11-s − 8.44·12-s − 11.0·14-s + 2.22·15-s + 13.3·16-s − 4.19·17-s − 1.21·18-s − 3.06·19-s − 7.37·20-s + 6.51·21-s + 2.69·22-s − 4.66·23-s − 14.1·24-s − 3.05·25-s + 5.50·27-s − 21.5·28-s + 1.18·29-s + 6.01·30-s + ⋯ |
L(s) = 1 | + 1.90·2-s − 0.921·3-s + 2.64·4-s − 0.624·5-s − 1.75·6-s − 1.54·7-s + 3.13·8-s − 0.150·9-s − 1.19·10-s + 0.301·11-s − 2.43·12-s − 2.94·14-s + 0.575·15-s + 3.34·16-s − 1.01·17-s − 0.286·18-s − 0.704·19-s − 1.64·20-s + 1.42·21-s + 0.575·22-s − 0.971·23-s − 2.89·24-s − 0.610·25-s + 1.06·27-s − 4.07·28-s + 0.219·29-s + 1.09·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - 2.69T + 2T^{2} \) |
| 3 | \( 1 + 1.59T + 3T^{2} \) |
| 5 | \( 1 + 1.39T + 5T^{2} \) |
| 7 | \( 1 + 4.07T + 7T^{2} \) |
| 17 | \( 1 + 4.19T + 17T^{2} \) |
| 19 | \( 1 + 3.06T + 19T^{2} \) |
| 23 | \( 1 + 4.66T + 23T^{2} \) |
| 29 | \( 1 - 1.18T + 29T^{2} \) |
| 31 | \( 1 + 4.60T + 31T^{2} \) |
| 37 | \( 1 + 2.26T + 37T^{2} \) |
| 41 | \( 1 + 6.57T + 41T^{2} \) |
| 43 | \( 1 - 4.36T + 43T^{2} \) |
| 47 | \( 1 + 4.67T + 47T^{2} \) |
| 53 | \( 1 - 3.57T + 53T^{2} \) |
| 59 | \( 1 + 2.70T + 59T^{2} \) |
| 61 | \( 1 + 8.11T + 61T^{2} \) |
| 67 | \( 1 - 15.3T + 67T^{2} \) |
| 71 | \( 1 + 10.7T + 71T^{2} \) |
| 73 | \( 1 + 10.7T + 73T^{2} \) |
| 79 | \( 1 - 4.42T + 79T^{2} \) |
| 83 | \( 1 - 11.5T + 83T^{2} \) |
| 89 | \( 1 - 8.50T + 89T^{2} \) |
| 97 | \( 1 - 4.35T + 97T^{2} \) |
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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
−8.796199639006015212921460128581, −7.56336890430159628479651230189, −6.65187300500264836890832224975, −6.32091345364534579507094377177, −5.66700105498633460148923362087, −4.69038884801844690837015765814, −3.91076922694115319375972908993, −3.25758743422242380489390487043, −2.15828069395988596741666183533, 0,
2.15828069395988596741666183533, 3.25758743422242380489390487043, 3.91076922694115319375972908993, 4.69038884801844690837015765814, 5.66700105498633460148923362087, 6.32091345364534579507094377177, 6.65187300500264836890832224975, 7.56336890430159628479651230189, 8.796199639006015212921460128581