| L(s) = 1 | − 2-s + 1.38·3-s + 4-s + 5-s − 1.38·6-s − 5.08·7-s − 8-s − 1.07·9-s − 10-s − 11-s + 1.38·12-s + 1.66·13-s + 5.08·14-s + 1.38·15-s + 16-s + 5.20·17-s + 1.07·18-s − 1.47·19-s + 20-s − 7.05·21-s + 22-s − 4.67·23-s − 1.38·24-s + 25-s − 1.66·26-s − 5.65·27-s − 5.08·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.801·3-s + 0.5·4-s + 0.447·5-s − 0.566·6-s − 1.92·7-s − 0.353·8-s − 0.358·9-s − 0.316·10-s − 0.301·11-s + 0.400·12-s + 0.462·13-s + 1.35·14-s + 0.358·15-s + 0.250·16-s + 1.26·17-s + 0.253·18-s − 0.338·19-s + 0.223·20-s − 1.53·21-s + 0.213·22-s − 0.974·23-s − 0.283·24-s + 0.200·25-s − 0.327·26-s − 1.08·27-s − 0.961·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.175600239\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.175600239\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 + T \) |
| good | 3 | \( 1 - 1.38T + 3T^{2} \) |
| 7 | \( 1 + 5.08T + 7T^{2} \) |
| 13 | \( 1 - 1.66T + 13T^{2} \) |
| 17 | \( 1 - 5.20T + 17T^{2} \) |
| 19 | \( 1 + 1.47T + 19T^{2} \) |
| 23 | \( 1 + 4.67T + 23T^{2} \) |
| 29 | \( 1 - 2.22T + 29T^{2} \) |
| 31 | \( 1 + 0.166T + 31T^{2} \) |
| 37 | \( 1 + 2.48T + 37T^{2} \) |
| 41 | \( 1 + 7.92T + 41T^{2} \) |
| 43 | \( 1 - 10.6T + 43T^{2} \) |
| 47 | \( 1 + 8.54T + 47T^{2} \) |
| 53 | \( 1 - 5.02T + 53T^{2} \) |
| 59 | \( 1 + 4.21T + 59T^{2} \) |
| 61 | \( 1 + 3.29T + 61T^{2} \) |
| 67 | \( 1 + 4.32T + 67T^{2} \) |
| 71 | \( 1 - 1.27T + 71T^{2} \) |
| 79 | \( 1 - 5.14T + 79T^{2} \) |
| 83 | \( 1 - 15.5T + 83T^{2} \) |
| 89 | \( 1 + 6.18T + 89T^{2} \) |
| 97 | \( 1 - 7.35T + 97T^{2} \) |
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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
−7.87639788020084434882233348970, −7.32979850316386322156941559160, −6.28599297867805378233636932529, −6.15511924754420668493701260723, −5.27470010064251860485887174243, −3.84852357871610052575127259711, −3.28186324612932499708046681968, −2.74207482717901083870276991185, −1.86965621916115403003761874424, −0.54558088398623928668553438479,
0.54558088398623928668553438479, 1.86965621916115403003761874424, 2.74207482717901083870276991185, 3.28186324612932499708046681968, 3.84852357871610052575127259711, 5.27470010064251860485887174243, 6.15511924754420668493701260723, 6.28599297867805378233636932529, 7.32979850316386322156941559160, 7.87639788020084434882233348970
