| L(s) = 1 | − 1.29·3-s − 3.79·5-s + 2.99·7-s − 1.32·9-s + 2.34·11-s + 13-s + 4.90·15-s + 1.52·17-s + 5.79·19-s − 3.87·21-s − 4.28·23-s + 9.37·25-s + 5.59·27-s − 29-s − 0.366·31-s − 3.03·33-s − 11.3·35-s + 3.94·37-s − 1.29·39-s − 4.70·41-s − 1.35·43-s + 5.03·45-s − 11.2·47-s + 1.97·49-s − 1.97·51-s + 6.22·53-s − 8.88·55-s + ⋯ |
| L(s) = 1 | − 0.746·3-s − 1.69·5-s + 1.13·7-s − 0.442·9-s + 0.706·11-s + 0.277·13-s + 1.26·15-s + 0.369·17-s + 1.32·19-s − 0.845·21-s − 0.892·23-s + 1.87·25-s + 1.07·27-s − 0.185·29-s − 0.0657·31-s − 0.527·33-s − 1.92·35-s + 0.648·37-s − 0.207·39-s − 0.734·41-s − 0.206·43-s + 0.750·45-s − 1.64·47-s + 0.282·49-s − 0.276·51-s + 0.855·53-s − 1.19·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.049561138\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.049561138\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 + 1.29T + 3T^{2} \) |
| 5 | \( 1 + 3.79T + 5T^{2} \) |
| 7 | \( 1 - 2.99T + 7T^{2} \) |
| 11 | \( 1 - 2.34T + 11T^{2} \) |
| 17 | \( 1 - 1.52T + 17T^{2} \) |
| 19 | \( 1 - 5.79T + 19T^{2} \) |
| 23 | \( 1 + 4.28T + 23T^{2} \) |
| 31 | \( 1 + 0.366T + 31T^{2} \) |
| 37 | \( 1 - 3.94T + 37T^{2} \) |
| 41 | \( 1 + 4.70T + 41T^{2} \) |
| 43 | \( 1 + 1.35T + 43T^{2} \) |
| 47 | \( 1 + 11.2T + 47T^{2} \) |
| 53 | \( 1 - 6.22T + 53T^{2} \) |
| 59 | \( 1 - 9.59T + 59T^{2} \) |
| 61 | \( 1 + 0.912T + 61T^{2} \) |
| 67 | \( 1 + 2.96T + 67T^{2} \) |
| 71 | \( 1 + 12.6T + 71T^{2} \) |
| 73 | \( 1 - 1.27T + 73T^{2} \) |
| 79 | \( 1 - 4.00T + 79T^{2} \) |
| 83 | \( 1 - 0.0710T + 83T^{2} \) |
| 89 | \( 1 + 2.95T + 89T^{2} \) |
| 97 | \( 1 + 0.0872T + 97T^{2} \) |
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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
−8.179813731618558226064470169349, −7.42474375483550345946309208352, −6.77892258775252082660789182664, −5.83502429116480165256559154926, −5.13812410184373977491651825404, −4.48094304666715770880781509061, −3.75467535528886875995055383832, −3.03028319074248917200688238908, −1.54430711618312405692061611357, −0.59392175364103712650914318550,
0.59392175364103712650914318550, 1.54430711618312405692061611357, 3.03028319074248917200688238908, 3.75467535528886875995055383832, 4.48094304666715770880781509061, 5.13812410184373977491651825404, 5.83502429116480165256559154926, 6.77892258775252082660789182664, 7.42474375483550345946309208352, 8.179813731618558226064470169349
