L(s) = 1 | + 0.404·2-s − 1.99·3-s − 1.83·4-s + 5-s − 0.805·6-s − 2.54·7-s − 1.55·8-s + 0.968·9-s + 0.404·10-s − 3.71·11-s + 3.65·12-s − 4.88·13-s − 1.02·14-s − 1.99·15-s + 3.04·16-s + 17-s + 0.391·18-s − 2.52·19-s − 1.83·20-s + 5.06·21-s − 1.50·22-s + 5.94·23-s + 3.08·24-s + 25-s − 1.97·26-s + 4.04·27-s + 4.66·28-s + ⋯ |
L(s) = 1 | + 0.285·2-s − 1.15·3-s − 0.918·4-s + 0.447·5-s − 0.328·6-s − 0.960·7-s − 0.548·8-s + 0.322·9-s + 0.127·10-s − 1.11·11-s + 1.05·12-s − 1.35·13-s − 0.274·14-s − 0.514·15-s + 0.761·16-s + 0.242·17-s + 0.0922·18-s − 0.578·19-s − 0.410·20-s + 1.10·21-s − 0.319·22-s + 1.23·23-s + 0.630·24-s + 0.200·25-s − 0.387·26-s + 0.778·27-s + 0.882·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 71 | \( 1 + T \) |
good | 2 | \( 1 - 0.404T + 2T^{2} \) |
| 3 | \( 1 + 1.99T + 3T^{2} \) |
| 7 | \( 1 + 2.54T + 7T^{2} \) |
| 11 | \( 1 + 3.71T + 11T^{2} \) |
| 13 | \( 1 + 4.88T + 13T^{2} \) |
| 19 | \( 1 + 2.52T + 19T^{2} \) |
| 23 | \( 1 - 5.94T + 23T^{2} \) |
| 29 | \( 1 - 9.14T + 29T^{2} \) |
| 31 | \( 1 - 9.91T + 31T^{2} \) |
| 37 | \( 1 + 1.61T + 37T^{2} \) |
| 41 | \( 1 + 3.99T + 41T^{2} \) |
| 43 | \( 1 - 5.28T + 43T^{2} \) |
| 47 | \( 1 + 8.72T + 47T^{2} \) |
| 53 | \( 1 - 8.60T + 53T^{2} \) |
| 59 | \( 1 + 0.753T + 59T^{2} \) |
| 61 | \( 1 + 13.1T + 61T^{2} \) |
| 67 | \( 1 + 2.42T + 67T^{2} \) |
| 73 | \( 1 - 12.7T + 73T^{2} \) |
| 79 | \( 1 - 9.28T + 79T^{2} \) |
| 83 | \( 1 - 7.27T + 83T^{2} \) |
| 89 | \( 1 - 1.63T + 89T^{2} \) |
| 97 | \( 1 - 1.68T + 97T^{2} \) |
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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
−7.69994962471070853745212324050, −6.54825546504459574868135285019, −6.39845632459549251295694715936, −5.29728290330925721515045135003, −5.05181480351169707635531130435, −4.42781969728380576345798946393, −3.07337376413837237228200490643, −2.64666105972537806919426349420, −0.851526624198747068817892316293, 0,
0.851526624198747068817892316293, 2.64666105972537806919426349420, 3.07337376413837237228200490643, 4.42781969728380576345798946393, 5.05181480351169707635531130435, 5.29728290330925721515045135003, 6.39845632459549251295694715936, 6.54825546504459574868135285019, 7.69994962471070853745212324050