| L(s) = 1 | + 2.66·2-s + 5.11·4-s + 1.38·5-s + 0.919·7-s + 8.32·8-s + 3.69·10-s + 4.03·11-s − 2.97·13-s + 2.45·14-s + 11.9·16-s − 0.220·17-s − 1.62·19-s + 7.07·20-s + 10.7·22-s − 23-s − 3.08·25-s − 7.94·26-s + 4.70·28-s + 29-s + 6.44·31-s + 15.2·32-s − 0.587·34-s + 1.27·35-s − 5.53·37-s − 4.34·38-s + 11.5·40-s + 3.81·41-s + ⋯ |
| L(s) = 1 | + 1.88·2-s + 2.55·4-s + 0.618·5-s + 0.347·7-s + 2.94·8-s + 1.16·10-s + 1.21·11-s − 0.826·13-s + 0.655·14-s + 2.99·16-s − 0.0534·17-s − 0.373·19-s + 1.58·20-s + 2.29·22-s − 0.208·23-s − 0.617·25-s − 1.55·26-s + 0.889·28-s + 0.185·29-s + 1.15·31-s + 2.70·32-s − 0.100·34-s + 0.214·35-s − 0.910·37-s − 0.705·38-s + 1.81·40-s + 0.595·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.893337558\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.893337558\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - 2.66T + 2T^{2} \) |
| 5 | \( 1 - 1.38T + 5T^{2} \) |
| 7 | \( 1 - 0.919T + 7T^{2} \) |
| 11 | \( 1 - 4.03T + 11T^{2} \) |
| 13 | \( 1 + 2.97T + 13T^{2} \) |
| 17 | \( 1 + 0.220T + 17T^{2} \) |
| 19 | \( 1 + 1.62T + 19T^{2} \) |
| 31 | \( 1 - 6.44T + 31T^{2} \) |
| 37 | \( 1 + 5.53T + 37T^{2} \) |
| 41 | \( 1 - 3.81T + 41T^{2} \) |
| 43 | \( 1 - 10.8T + 43T^{2} \) |
| 47 | \( 1 - 10.2T + 47T^{2} \) |
| 53 | \( 1 + 2.84T + 53T^{2} \) |
| 59 | \( 1 - 7.16T + 59T^{2} \) |
| 61 | \( 1 + 0.284T + 61T^{2} \) |
| 67 | \( 1 - 4.27T + 67T^{2} \) |
| 71 | \( 1 + 2.88T + 71T^{2} \) |
| 73 | \( 1 + 12.4T + 73T^{2} \) |
| 79 | \( 1 + 12.0T + 79T^{2} \) |
| 83 | \( 1 - 0.373T + 83T^{2} \) |
| 89 | \( 1 + 7.58T + 89T^{2} \) |
| 97 | \( 1 + 1.60T + 97T^{2} \) |
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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
−7.69655409987559240381790842723, −7.06299620147930057162641013494, −6.37274434120428799112334580976, −5.85688411685526001843003210272, −5.17420636567725339883351436398, −4.33104727104565581155581093571, −3.98314070108358042306390898664, −2.86005489811600686551140250758, −2.21471256911562383908027540399, −1.34420591397500087005142675514,
1.34420591397500087005142675514, 2.21471256911562383908027540399, 2.86005489811600686551140250758, 3.98314070108358042306390898664, 4.33104727104565581155581093571, 5.17420636567725339883351436398, 5.85688411685526001843003210272, 6.37274434120428799112334580976, 7.06299620147930057162641013494, 7.69655409987559240381790842723
