L(s) = 1 | + 2-s + 4-s − 5-s + 8-s − 10-s + 6.24·11-s − 5.65·13-s + 16-s − 5.41·17-s − 1.17·19-s − 20-s + 6.24·22-s − 8.82·23-s + 25-s − 5.65·26-s − 5.41·29-s − 1.75·31-s + 32-s − 5.41·34-s + 8.24·37-s − 1.17·38-s − 40-s + 6.48·41-s − 5.07·43-s + 6.24·44-s − 8.82·46-s − 6.24·47-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.447·5-s + 0.353·8-s − 0.316·10-s + 1.88·11-s − 1.56·13-s + 0.250·16-s − 1.31·17-s − 0.268·19-s − 0.223·20-s + 1.33·22-s − 1.84·23-s + 0.200·25-s − 1.10·26-s − 1.00·29-s − 0.315·31-s + 0.176·32-s − 0.928·34-s + 1.35·37-s − 0.190·38-s − 0.158·40-s + 1.01·41-s − 0.773·43-s + 0.941·44-s − 1.30·46-s − 0.910·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 6.24T + 11T^{2} \) |
| 13 | \( 1 + 5.65T + 13T^{2} \) |
| 17 | \( 1 + 5.41T + 17T^{2} \) |
| 19 | \( 1 + 1.17T + 19T^{2} \) |
| 23 | \( 1 + 8.82T + 23T^{2} \) |
| 29 | \( 1 + 5.41T + 29T^{2} \) |
| 31 | \( 1 + 1.75T + 31T^{2} \) |
| 37 | \( 1 - 8.24T + 37T^{2} \) |
| 41 | \( 1 - 6.48T + 41T^{2} \) |
| 43 | \( 1 + 5.07T + 43T^{2} \) |
| 47 | \( 1 + 6.24T + 47T^{2} \) |
| 53 | \( 1 + 11.6T + 53T^{2} \) |
| 59 | \( 1 - 8.82T + 59T^{2} \) |
| 61 | \( 1 + 0.343T + 61T^{2} \) |
| 67 | \( 1 + 5.07T + 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 10.8T + 83T^{2} \) |
| 89 | \( 1 + 4.82T + 89T^{2} \) |
| 97 | \( 1 - 10.4T + 97T^{2} \) |
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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
−7.79841553246896125315929623793, −7.17938601771480336338588393461, −6.44581788381212753953679251893, −5.92295703756200624851977470277, −4.71403504813075061615957161966, −4.27779640733829878575313115650, −3.61779790633090000651009274081, −2.45985048022650625306439424211, −1.66959896042508857476113831357, 0,
1.66959896042508857476113831357, 2.45985048022650625306439424211, 3.61779790633090000651009274081, 4.27779640733829878575313115650, 4.71403504813075061615957161966, 5.92295703756200624851977470277, 6.44581788381212753953679251893, 7.17938601771480336338588393461, 7.79841553246896125315929623793