| L(s) = 1 | − 2.21·2-s + 0.921·3-s + 2.89·4-s − 5-s − 2.03·6-s − 4.71·7-s − 1.99·8-s − 2.15·9-s + 2.21·10-s − 11-s + 2.67·12-s − 1.04·13-s + 10.4·14-s − 0.921·15-s − 1.39·16-s − 5.55·17-s + 4.76·18-s + 19-s − 2.89·20-s − 4.34·21-s + 2.21·22-s + 6.35·23-s − 1.83·24-s + 25-s + 2.30·26-s − 4.74·27-s − 13.6·28-s + ⋯ |
| L(s) = 1 | − 1.56·2-s + 0.531·3-s + 1.44·4-s − 0.447·5-s − 0.832·6-s − 1.78·7-s − 0.703·8-s − 0.717·9-s + 0.699·10-s − 0.301·11-s + 0.771·12-s − 0.289·13-s + 2.79·14-s − 0.237·15-s − 0.348·16-s − 1.34·17-s + 1.12·18-s + 0.229·19-s − 0.648·20-s − 0.948·21-s + 0.471·22-s + 1.32·23-s − 0.374·24-s + 0.200·25-s + 0.452·26-s − 0.913·27-s − 2.58·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4040600920\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4040600920\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + 2.21T + 2T^{2} \) |
| 3 | \( 1 - 0.921T + 3T^{2} \) |
| 7 | \( 1 + 4.71T + 7T^{2} \) |
| 13 | \( 1 + 1.04T + 13T^{2} \) |
| 17 | \( 1 + 5.55T + 17T^{2} \) |
| 23 | \( 1 - 6.35T + 23T^{2} \) |
| 29 | \( 1 - 4.73T + 29T^{2} \) |
| 31 | \( 1 - 8.01T + 31T^{2} \) |
| 37 | \( 1 - 5.19T + 37T^{2} \) |
| 41 | \( 1 - 7.09T + 41T^{2} \) |
| 43 | \( 1 + 1.79T + 43T^{2} \) |
| 47 | \( 1 + 9.17T + 47T^{2} \) |
| 53 | \( 1 - 5.94T + 53T^{2} \) |
| 59 | \( 1 - 4.81T + 59T^{2} \) |
| 61 | \( 1 + 6.30T + 61T^{2} \) |
| 67 | \( 1 + 8.73T + 67T^{2} \) |
| 71 | \( 1 - 5.74T + 71T^{2} \) |
| 73 | \( 1 + 15.5T + 73T^{2} \) |
| 79 | \( 1 + 11.8T + 79T^{2} \) |
| 83 | \( 1 - 16.5T + 83T^{2} \) |
| 89 | \( 1 - 8.58T + 89T^{2} \) |
| 97 | \( 1 - 9.73T + 97T^{2} \) |
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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
−9.696563198343117826545989981225, −9.037476366746965163316334818879, −8.553518865668267649148389895563, −7.62327732165337906808407607230, −6.82030006698688382932408355428, −6.16528264577717189841345436221, −4.54782719169401016434638787678, −3.09480418827593333203030152411, −2.54353610244633251072533367277, −0.56582160983548748092752148829,
0.56582160983548748092752148829, 2.54353610244633251072533367277, 3.09480418827593333203030152411, 4.54782719169401016434638787678, 6.16528264577717189841345436221, 6.82030006698688382932408355428, 7.62327732165337906808407607230, 8.553518865668267649148389895563, 9.037476366746965163316334818879, 9.696563198343117826545989981225
