L(s) = 1 | − 2.59·3-s + 0.486·5-s + 7-s + 3.73·9-s + 11-s + 13-s − 1.26·15-s + 4.76·17-s − 5.64·19-s − 2.59·21-s − 5.89·23-s − 4.76·25-s − 1.90·27-s + 5.02·29-s + 7.10·31-s − 2.59·33-s + 0.486·35-s + 3.80·37-s − 2.59·39-s − 5.77·41-s − 1.96·43-s + 1.81·45-s − 3.82·47-s + 49-s − 12.3·51-s + 1.43·53-s + 0.486·55-s + ⋯ |
L(s) = 1 | − 1.49·3-s + 0.217·5-s + 0.377·7-s + 1.24·9-s + 0.301·11-s + 0.277·13-s − 0.326·15-s + 1.15·17-s − 1.29·19-s − 0.566·21-s − 1.22·23-s − 0.952·25-s − 0.367·27-s + 0.932·29-s + 1.27·31-s − 0.451·33-s + 0.0823·35-s + 0.626·37-s − 0.415·39-s − 0.902·41-s − 0.299·43-s + 0.271·45-s − 0.558·47-s + 0.142·49-s − 1.73·51-s + 0.197·53-s + 0.0656·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
good | 3 | \( 1 + 2.59T + 3T^{2} \) |
| 5 | \( 1 - 0.486T + 5T^{2} \) |
| 17 | \( 1 - 4.76T + 17T^{2} \) |
| 19 | \( 1 + 5.64T + 19T^{2} \) |
| 23 | \( 1 + 5.89T + 23T^{2} \) |
| 29 | \( 1 - 5.02T + 29T^{2} \) |
| 31 | \( 1 - 7.10T + 31T^{2} \) |
| 37 | \( 1 - 3.80T + 37T^{2} \) |
| 41 | \( 1 + 5.77T + 41T^{2} \) |
| 43 | \( 1 + 1.96T + 43T^{2} \) |
| 47 | \( 1 + 3.82T + 47T^{2} \) |
| 53 | \( 1 - 1.43T + 53T^{2} \) |
| 59 | \( 1 + 5.08T + 59T^{2} \) |
| 61 | \( 1 + 11.4T + 61T^{2} \) |
| 67 | \( 1 - 7.40T + 67T^{2} \) |
| 71 | \( 1 + 16.5T + 71T^{2} \) |
| 73 | \( 1 + 10.6T + 73T^{2} \) |
| 79 | \( 1 - 4.42T + 79T^{2} \) |
| 83 | \( 1 + 10.8T + 83T^{2} \) |
| 89 | \( 1 - 14.7T + 89T^{2} \) |
| 97 | \( 1 - 9.76T + 97T^{2} \) |
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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
−7.41968509530975529253432367134, −6.38726199086536077407880714444, −6.18867599471717976756719255715, −5.56146156175283862299439203611, −4.66243915618972031239014961011, −4.26331715169386929041235931466, −3.16811059016404385167860921953, −1.95186565136203796041376299459, −1.12396990776620900929751655106, 0,
1.12396990776620900929751655106, 1.95186565136203796041376299459, 3.16811059016404385167860921953, 4.26331715169386929041235931466, 4.66243915618972031239014961011, 5.56146156175283862299439203611, 6.18867599471717976756719255715, 6.38726199086536077407880714444, 7.41968509530975529253432367134