L(s) = 1 | − 1.51·2-s + 5.95·3-s − 5.69·4-s − 9.03·6-s + 3.35·7-s + 20.7·8-s + 8.40·9-s + 20.2·11-s − 33.8·12-s + 5.57·13-s − 5.09·14-s + 13.9·16-s + 26.5·17-s − 12.7·18-s − 25.3·19-s + 19.9·21-s − 30.7·22-s + 90.2·23-s + 123.·24-s − 8.46·26-s − 110.·27-s − 19.1·28-s − 123.·29-s + 129.·31-s − 187.·32-s + 120.·33-s − 40.2·34-s + ⋯ |
L(s) = 1 | − 0.536·2-s + 1.14·3-s − 0.711·4-s − 0.614·6-s + 0.181·7-s + 0.919·8-s + 0.311·9-s + 0.554·11-s − 0.814·12-s + 0.118·13-s − 0.0973·14-s + 0.218·16-s + 0.378·17-s − 0.167·18-s − 0.306·19-s + 0.207·21-s − 0.297·22-s + 0.818·23-s + 1.05·24-s − 0.0638·26-s − 0.788·27-s − 0.128·28-s − 0.789·29-s + 0.750·31-s − 1.03·32-s + 0.634·33-s − 0.203·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.082544401\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.082544401\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 47 | \( 1 + 47T \) |
good | 2 | \( 1 + 1.51T + 8T^{2} \) |
| 3 | \( 1 - 5.95T + 27T^{2} \) |
| 7 | \( 1 - 3.35T + 343T^{2} \) |
| 11 | \( 1 - 20.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 5.57T + 2.19e3T^{2} \) |
| 17 | \( 1 - 26.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 25.3T + 6.85e3T^{2} \) |
| 23 | \( 1 - 90.2T + 1.21e4T^{2} \) |
| 29 | \( 1 + 123.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 129.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 213.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 124.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 424.T + 7.95e4T^{2} \) |
| 53 | \( 1 + 361.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 836.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 194.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 902.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 690.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 698.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 449.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 543.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 725.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 214.T + 9.12e5T^{2} \) |
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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
−9.273604700508178925567952752779, −8.661535341509424576294226178845, −7.994032970991170218435366373119, −7.33681484205522815136160844077, −6.10510903025516307938248923744, −4.94438638334739554779502450213, −4.02470683484685360097751601046, −3.17247929043010025150706433091, −1.94827046596921416600186491710, −0.789840490471609575049896819847,
0.789840490471609575049896819847, 1.94827046596921416600186491710, 3.17247929043010025150706433091, 4.02470683484685360097751601046, 4.94438638334739554779502450213, 6.10510903025516307938248923744, 7.33681484205522815136160844077, 7.994032970991170218435366373119, 8.661535341509424576294226178845, 9.273604700508178925567952752779