L(s) = 1 | − 2-s + 3-s + 4-s − 3.75·5-s − 6-s + 1.91·7-s − 8-s + 9-s + 3.75·10-s − 4.14·11-s + 12-s − 13-s − 1.91·14-s − 3.75·15-s + 16-s − 4.83·17-s − 18-s + 2.13·19-s − 3.75·20-s + 1.91·21-s + 4.14·22-s + 2.08·23-s − 24-s + 9.07·25-s + 26-s + 27-s + 1.91·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.67·5-s − 0.408·6-s + 0.723·7-s − 0.353·8-s + 0.333·9-s + 1.18·10-s − 1.25·11-s + 0.288·12-s − 0.277·13-s − 0.511·14-s − 0.968·15-s + 0.250·16-s − 1.17·17-s − 0.235·18-s + 0.489·19-s − 0.839·20-s + 0.417·21-s + 0.884·22-s + 0.434·23-s − 0.204·24-s + 1.81·25-s + 0.196·26-s + 0.192·27-s + 0.361·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
good | 5 | \( 1 + 3.75T + 5T^{2} \) |
| 7 | \( 1 - 1.91T + 7T^{2} \) |
| 11 | \( 1 + 4.14T + 11T^{2} \) |
| 17 | \( 1 + 4.83T + 17T^{2} \) |
| 19 | \( 1 - 2.13T + 19T^{2} \) |
| 23 | \( 1 - 2.08T + 23T^{2} \) |
| 29 | \( 1 + 2.19T + 29T^{2} \) |
| 31 | \( 1 - 5.15T + 31T^{2} \) |
| 37 | \( 1 - 9.63T + 37T^{2} \) |
| 41 | \( 1 - 7.88T + 41T^{2} \) |
| 43 | \( 1 - 1.60T + 43T^{2} \) |
| 47 | \( 1 + 9.35T + 47T^{2} \) |
| 53 | \( 1 - 12.8T + 53T^{2} \) |
| 59 | \( 1 + 7.21T + 59T^{2} \) |
| 61 | \( 1 - 10.2T + 61T^{2} \) |
| 67 | \( 1 + 8.07T + 67T^{2} \) |
| 71 | \( 1 - 9.46T + 71T^{2} \) |
| 73 | \( 1 - 4.11T + 73T^{2} \) |
| 79 | \( 1 + 8.08T + 79T^{2} \) |
| 83 | \( 1 - 0.397T + 83T^{2} \) |
| 89 | \( 1 + 12.6T + 89T^{2} \) |
| 97 | \( 1 + 14.9T + 97T^{2} \) |
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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
−7.58435117305629765212728042026, −7.29876670592252445234126574073, −6.37545026967720291694374991337, −5.19973095624475414292854070032, −4.55222128622469853462770188019, −3.90901128024172871775655960518, −2.88089691741642071754396156239, −2.38658854951526100383703985050, −1.04879240734906847598177560747, 0,
1.04879240734906847598177560747, 2.38658854951526100383703985050, 2.88089691741642071754396156239, 3.90901128024172871775655960518, 4.55222128622469853462770188019, 5.19973095624475414292854070032, 6.37545026967720291694374991337, 7.29876670592252445234126574073, 7.58435117305629765212728042026