L(s) = 1 | + 1.67·3-s − 5-s + 4.15·7-s − 0.193·9-s + 0.806·11-s − 0.481·13-s − 1.67·15-s − 3.61·17-s + 19-s + 6.96·21-s − 5.76·23-s + 25-s − 5.35·27-s + 8.57·29-s + 3.61·31-s + 1.35·33-s − 4.15·35-s + 9.18·37-s − 0.806·39-s + 0.312·41-s + 3.19·43-s + 0.193·45-s − 5.76·47-s + 10.2·49-s − 6.05·51-s + 8.21·53-s − 0.806·55-s + ⋯ |
L(s) = 1 | + 0.967·3-s − 0.447·5-s + 1.57·7-s − 0.0646·9-s + 0.243·11-s − 0.133·13-s − 0.432·15-s − 0.876·17-s + 0.229·19-s + 1.51·21-s − 1.20·23-s + 0.200·25-s − 1.02·27-s + 1.59·29-s + 0.648·31-s + 0.235·33-s − 0.702·35-s + 1.50·37-s − 0.129·39-s + 0.0488·41-s + 0.487·43-s + 0.0289·45-s − 0.841·47-s + 1.46·49-s − 0.847·51-s + 1.12·53-s − 0.108·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.105100144\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.105100144\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 - 1.67T + 3T^{2} \) |
| 7 | \( 1 - 4.15T + 7T^{2} \) |
| 11 | \( 1 - 0.806T + 11T^{2} \) |
| 13 | \( 1 + 0.481T + 13T^{2} \) |
| 17 | \( 1 + 3.61T + 17T^{2} \) |
| 23 | \( 1 + 5.76T + 23T^{2} \) |
| 29 | \( 1 - 8.57T + 29T^{2} \) |
| 31 | \( 1 - 3.61T + 31T^{2} \) |
| 37 | \( 1 - 9.18T + 37T^{2} \) |
| 41 | \( 1 - 0.312T + 41T^{2} \) |
| 43 | \( 1 - 3.19T + 43T^{2} \) |
| 47 | \( 1 + 5.76T + 47T^{2} \) |
| 53 | \( 1 - 8.21T + 53T^{2} \) |
| 59 | \( 1 - 8.88T + 59T^{2} \) |
| 61 | \( 1 - 5.58T + 61T^{2} \) |
| 67 | \( 1 + 9.59T + 67T^{2} \) |
| 71 | \( 1 - 16.2T + 71T^{2} \) |
| 73 | \( 1 + 2.12T + 73T^{2} \) |
| 79 | \( 1 - 16.8T + 79T^{2} \) |
| 83 | \( 1 - 5.89T + 83T^{2} \) |
| 89 | \( 1 + 11.2T + 89T^{2} \) |
| 97 | \( 1 - 17.4T + 97T^{2} \) |
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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
−8.167563300927790231540668472037, −7.70089125928789776483985215002, −6.79731248425762618291820850515, −5.95118314181983831190784966385, −4.96871982537810166180751571418, −4.38082524438121947896708005647, −3.71260367545991939151686305059, −2.59357150607033699117566629479, −2.08684099872866266532992437071, −0.899616230530764158405761609521,
0.899616230530764158405761609521, 2.08684099872866266532992437071, 2.59357150607033699117566629479, 3.71260367545991939151686305059, 4.38082524438121947896708005647, 4.96871982537810166180751571418, 5.95118314181983831190784966385, 6.79731248425762618291820850515, 7.70089125928789776483985215002, 8.167563300927790231540668472037