L(s) = 1 | − 2-s + 0.408·3-s + 4-s − 5-s − 0.408·6-s + 2.53·7-s − 8-s − 2.83·9-s + 10-s − 1.03·11-s + 0.408·12-s + 4.74·13-s − 2.53·14-s − 0.408·15-s + 16-s + 2.42·17-s + 2.83·18-s − 6.59·19-s − 20-s + 1.03·21-s + 1.03·22-s − 0.408·24-s + 25-s − 4.74·26-s − 2.38·27-s + 2.53·28-s − 10.1·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.235·3-s + 0.5·4-s − 0.447·5-s − 0.166·6-s + 0.956·7-s − 0.353·8-s − 0.944·9-s + 0.316·10-s − 0.310·11-s + 0.117·12-s + 1.31·13-s − 0.676·14-s − 0.105·15-s + 0.250·16-s + 0.588·17-s + 0.667·18-s − 1.51·19-s − 0.223·20-s + 0.225·21-s + 0.219·22-s − 0.0833·24-s + 0.200·25-s − 0.931·26-s − 0.458·27-s + 0.478·28-s − 1.87·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 \) |
good | 3 | \( 1 - 0.408T + 3T^{2} \) |
| 7 | \( 1 - 2.53T + 7T^{2} \) |
| 11 | \( 1 + 1.03T + 11T^{2} \) |
| 13 | \( 1 - 4.74T + 13T^{2} \) |
| 17 | \( 1 - 2.42T + 17T^{2} \) |
| 19 | \( 1 + 6.59T + 19T^{2} \) |
| 29 | \( 1 + 10.1T + 29T^{2} \) |
| 31 | \( 1 - 3.03T + 31T^{2} \) |
| 37 | \( 1 - 11.6T + 37T^{2} \) |
| 41 | \( 1 + 6.55T + 41T^{2} \) |
| 43 | \( 1 - 1.05T + 43T^{2} \) |
| 47 | \( 1 - 2.53T + 47T^{2} \) |
| 53 | \( 1 + 8.59T + 53T^{2} \) |
| 59 | \( 1 + 11.7T + 59T^{2} \) |
| 61 | \( 1 - 2.19T + 61T^{2} \) |
| 67 | \( 1 + 9.87T + 67T^{2} \) |
| 71 | \( 1 + 3.37T + 71T^{2} \) |
| 73 | \( 1 - 14.7T + 73T^{2} \) |
| 79 | \( 1 + 4.32T + 79T^{2} \) |
| 83 | \( 1 + 3.62T + 83T^{2} \) |
| 89 | \( 1 - 5.83T + 89T^{2} \) |
| 97 | \( 1 + 1.77T + 97T^{2} \) |
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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
−7.972738772798789771082890012361, −7.51349795949819075644962749284, −6.30622864651898476743564790474, −5.91107987274018928915846636771, −4.91247689426930594051177474992, −4.00047805847999920101711817267, −3.19725245363441420763248342605, −2.22046346707280676142197658618, −1.32396481504076558411394422468, 0,
1.32396481504076558411394422468, 2.22046346707280676142197658618, 3.19725245363441420763248342605, 4.00047805847999920101711817267, 4.91247689426930594051177474992, 5.91107987274018928915846636771, 6.30622864651898476743564790474, 7.51349795949819075644962749284, 7.972738772798789771082890012361