| L(s) = 1 | + 2.40·2-s + 3-s + 3.78·4-s + 5-s + 2.40·6-s + 3.14·7-s + 4.29·8-s + 9-s + 2.40·10-s + 3.52·11-s + 3.78·12-s + 2.52·13-s + 7.57·14-s + 15-s + 2.76·16-s − 0.405·17-s + 2.40·18-s + 3.12·19-s + 3.78·20-s + 3.14·21-s + 8.48·22-s + 4.29·24-s + 25-s + 6.08·26-s + 27-s + 11.9·28-s + 2·29-s + ⋯ |
| L(s) = 1 | + 1.70·2-s + 0.577·3-s + 1.89·4-s + 0.447·5-s + 0.981·6-s + 1.18·7-s + 1.51·8-s + 0.333·9-s + 0.760·10-s + 1.06·11-s + 1.09·12-s + 0.701·13-s + 2.02·14-s + 0.258·15-s + 0.690·16-s − 0.0983·17-s + 0.566·18-s + 0.716·19-s + 0.846·20-s + 0.686·21-s + 1.80·22-s + 0.876·24-s + 0.200·25-s + 1.19·26-s + 0.192·27-s + 2.25·28-s + 0.371·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.36505716\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.36505716\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 - 2.40T + 2T^{2} \) |
| 7 | \( 1 - 3.14T + 7T^{2} \) |
| 11 | \( 1 - 3.52T + 11T^{2} \) |
| 13 | \( 1 - 2.52T + 13T^{2} \) |
| 17 | \( 1 + 0.405T + 17T^{2} \) |
| 19 | \( 1 - 3.12T + 19T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 6.23T + 31T^{2} \) |
| 37 | \( 1 + 2.01T + 37T^{2} \) |
| 41 | \( 1 + 12.4T + 41T^{2} \) |
| 43 | \( 1 + 12.3T + 43T^{2} \) |
| 47 | \( 1 + 7.86T + 47T^{2} \) |
| 53 | \( 1 - 13.2T + 53T^{2} \) |
| 59 | \( 1 + 11.9T + 59T^{2} \) |
| 61 | \( 1 - 6.11T + 61T^{2} \) |
| 67 | \( 1 - 1.14T + 67T^{2} \) |
| 71 | \( 1 - 1.52T + 71T^{2} \) |
| 73 | \( 1 - 2.51T + 73T^{2} \) |
| 79 | \( 1 + 14.1T + 79T^{2} \) |
| 83 | \( 1 - 7.97T + 83T^{2} \) |
| 89 | \( 1 + 2.21T + 89T^{2} \) |
| 97 | \( 1 - 6.32T + 97T^{2} \) |
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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
−7.67527931843629256191764827161, −6.78780341544043941507338180038, −6.48352198412118650661903146505, −5.38657061041788146323989923418, −5.11776613694172644911826657911, −4.24114233352824475339140553334, −3.62199627613879074103470995541, −3.00315395555000909756515612886, −1.82407560806953277706676184913, −1.51374979075763390456753738050,
1.51374979075763390456753738050, 1.82407560806953277706676184913, 3.00315395555000909756515612886, 3.62199627613879074103470995541, 4.24114233352824475339140553334, 5.11776613694172644911826657911, 5.38657061041788146323989923418, 6.48352198412118650661903146505, 6.78780341544043941507338180038, 7.67527931843629256191764827161
