L(s) = 1 | + 2.82·3-s − 5-s + 5.00·9-s + 2.82·11-s − 13-s − 2.82·15-s + 2·17-s + 2.82·19-s + 2.82·23-s + 25-s + 5.65·27-s + 2·29-s + 2.82·31-s + 8.00·33-s + 2·37-s − 2.82·39-s − 6·41-s − 2.82·43-s − 5.00·45-s − 7·49-s + 5.65·51-s − 6·53-s − 2.82·55-s + 8.00·57-s + 8.48·59-s + 2·61-s + 65-s + ⋯ |
L(s) = 1 | + 1.63·3-s − 0.447·5-s + 1.66·9-s + 0.852·11-s − 0.277·13-s − 0.730·15-s + 0.485·17-s + 0.648·19-s + 0.589·23-s + 0.200·25-s + 1.08·27-s + 0.371·29-s + 0.508·31-s + 1.39·33-s + 0.328·37-s − 0.452·39-s − 0.937·41-s − 0.431·43-s − 0.745·45-s − 49-s + 0.792·51-s − 0.824·53-s − 0.381·55-s + 1.05·57-s + 1.10·59-s + 0.256·61-s + 0.124·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.665892305\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.665892305\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
good | 3 | \( 1 - 2.82T + 3T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 - 2.82T + 11T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 - 2.82T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 2.82T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 2.82T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 8.48T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 5.65T + 67T^{2} \) |
| 71 | \( 1 - 14.1T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 + 5.65T + 79T^{2} \) |
| 83 | \( 1 + 11.3T + 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
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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
−8.296754070010013689876886288530, −7.944761334529846490528994091005, −7.08587596106090714501750327432, −6.54652074337568047902675340697, −5.25839021985048106488659128635, −4.42226619311924558214904423108, −3.55567408189188797034868749768, −3.11637133213322795089069338279, −2.11478842125994345173605950595, −1.07648691254469932758360495889,
1.07648691254469932758360495889, 2.11478842125994345173605950595, 3.11637133213322795089069338279, 3.55567408189188797034868749768, 4.42226619311924558214904423108, 5.25839021985048106488659128635, 6.54652074337568047902675340697, 7.08587596106090714501750327432, 7.944761334529846490528994091005, 8.296754070010013689876886288530