L(s) = 1 | − 1.35·2-s − 0.162·4-s + 5-s − 3.64·7-s + 2.93·8-s − 1.35·10-s − 2.83·13-s + 4.94·14-s − 3.64·16-s − 3.69·17-s − 0.0951·19-s − 0.162·20-s − 1.16·23-s + 25-s + 3.83·26-s + 0.591·28-s + 6.75·29-s + 6.77·31-s − 0.914·32-s + 5.00·34-s − 3.64·35-s + 9.83·37-s + 0.128·38-s + 2.93·40-s + 8.31·41-s − 2.96·43-s + 1.57·46-s + ⋯ |
L(s) = 1 | − 0.958·2-s − 0.0810·4-s + 0.447·5-s − 1.37·7-s + 1.03·8-s − 0.428·10-s − 0.785·13-s + 1.32·14-s − 0.912·16-s − 0.895·17-s − 0.0218·19-s − 0.0362·20-s − 0.242·23-s + 0.200·25-s + 0.752·26-s + 0.111·28-s + 1.25·29-s + 1.21·31-s − 0.161·32-s + 0.858·34-s − 0.616·35-s + 1.61·37-s + 0.0209·38-s + 0.463·40-s + 1.29·41-s − 0.452·43-s + 0.232·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + 1.35T + 2T^{2} \) |
| 7 | \( 1 + 3.64T + 7T^{2} \) |
| 13 | \( 1 + 2.83T + 13T^{2} \) |
| 17 | \( 1 + 3.69T + 17T^{2} \) |
| 19 | \( 1 + 0.0951T + 19T^{2} \) |
| 23 | \( 1 + 1.16T + 23T^{2} \) |
| 29 | \( 1 - 6.75T + 29T^{2} \) |
| 31 | \( 1 - 6.77T + 31T^{2} \) |
| 37 | \( 1 - 9.83T + 37T^{2} \) |
| 41 | \( 1 - 8.31T + 41T^{2} \) |
| 43 | \( 1 + 2.96T + 43T^{2} \) |
| 47 | \( 1 - 2.22T + 47T^{2} \) |
| 53 | \( 1 + 2.99T + 53T^{2} \) |
| 59 | \( 1 - 8.50T + 59T^{2} \) |
| 61 | \( 1 + 8.48T + 61T^{2} \) |
| 67 | \( 1 + 13.4T + 67T^{2} \) |
| 71 | \( 1 + 8.30T + 71T^{2} \) |
| 73 | \( 1 + 1.32T + 73T^{2} \) |
| 79 | \( 1 + 13.8T + 79T^{2} \) |
| 83 | \( 1 - 10.6T + 83T^{2} \) |
| 89 | \( 1 - 12.1T + 89T^{2} \) |
| 97 | \( 1 + 4.33T + 97T^{2} \) |
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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
−7.899603583885823142487238473422, −7.18570614478777731686175811206, −6.46270764671842175988882361140, −5.92697210091665191598725503928, −4.72301207883392371487380939029, −4.26071990054352459930152276522, −2.98258343218293954963616812830, −2.34285631463948532861335358908, −1.02595641029530689625875798968, 0,
1.02595641029530689625875798968, 2.34285631463948532861335358908, 2.98258343218293954963616812830, 4.26071990054352459930152276522, 4.72301207883392371487380939029, 5.92697210091665191598725503928, 6.46270764671842175988882361140, 7.18570614478777731686175811206, 7.899603583885823142487238473422