L(s) = 1 | + 2-s + 4-s + 1.77·5-s + 0.833·7-s + 8-s + 1.77·10-s + 3.92·11-s + 4.58·13-s + 0.833·14-s + 16-s + 1.16·17-s − 3.74·19-s + 1.77·20-s + 3.92·22-s − 1.83·25-s + 4.58·26-s + 0.833·28-s + 1.08·29-s + 7.05·31-s + 32-s + 1.16·34-s + 1.48·35-s + 5.44·37-s − 3.74·38-s + 1.77·40-s − 1.45·41-s + 3.91·43-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.795·5-s + 0.315·7-s + 0.353·8-s + 0.562·10-s + 1.18·11-s + 1.27·13-s + 0.222·14-s + 0.250·16-s + 0.282·17-s − 0.859·19-s + 0.397·20-s + 0.836·22-s − 0.366·25-s + 0.899·26-s + 0.157·28-s + 0.202·29-s + 1.26·31-s + 0.176·32-s + 0.200·34-s + 0.250·35-s + 0.895·37-s − 0.607·38-s + 0.281·40-s − 0.226·41-s + 0.597·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.148763855\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.148763855\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 23 | \( 1 \) |
good | 5 | \( 1 - 1.77T + 5T^{2} \) |
| 7 | \( 1 - 0.833T + 7T^{2} \) |
| 11 | \( 1 - 3.92T + 11T^{2} \) |
| 13 | \( 1 - 4.58T + 13T^{2} \) |
| 17 | \( 1 - 1.16T + 17T^{2} \) |
| 19 | \( 1 + 3.74T + 19T^{2} \) |
| 29 | \( 1 - 1.08T + 29T^{2} \) |
| 31 | \( 1 - 7.05T + 31T^{2} \) |
| 37 | \( 1 - 5.44T + 37T^{2} \) |
| 41 | \( 1 + 1.45T + 41T^{2} \) |
| 43 | \( 1 - 3.91T + 43T^{2} \) |
| 47 | \( 1 + 5.02T + 47T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 + 3.91T + 59T^{2} \) |
| 61 | \( 1 + 4.19T + 61T^{2} \) |
| 67 | \( 1 - 2.05T + 67T^{2} \) |
| 71 | \( 1 + 7.83T + 71T^{2} \) |
| 73 | \( 1 - 13.8T + 73T^{2} \) |
| 79 | \( 1 + 9.56T + 79T^{2} \) |
| 83 | \( 1 + 12.8T + 83T^{2} \) |
| 89 | \( 1 - 2.07T + 89T^{2} \) |
| 97 | \( 1 + 12.5T + 97T^{2} \) |
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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
−7.60747355180396215358899276174, −6.69723161558424941052694213645, −6.16142043958343368955248699562, −5.87838872011989375336650248321, −4.85831149046698737499467135651, −4.20148318947269267466722332013, −3.58401443433117883865320808370, −2.63992896819570323702270566402, −1.74605154950172393234741383120, −1.07526410127541234115853242738,
1.07526410127541234115853242738, 1.74605154950172393234741383120, 2.63992896819570323702270566402, 3.58401443433117883865320808370, 4.20148318947269267466722332013, 4.85831149046698737499467135651, 5.87838872011989375336650248321, 6.16142043958343368955248699562, 6.69723161558424941052694213645, 7.60747355180396215358899276174