L(s) = 1 | − 2-s + 0.839·3-s + 4-s + 3.67·5-s − 0.839·6-s − 0.282·7-s − 8-s − 2.29·9-s − 3.67·10-s − 0.500·11-s + 0.839·12-s − 6.43·13-s + 0.282·14-s + 3.08·15-s + 16-s − 4.42·17-s + 2.29·18-s + 6.17·19-s + 3.67·20-s − 0.237·21-s + 0.500·22-s − 23-s − 0.839·24-s + 8.52·25-s + 6.43·26-s − 4.44·27-s − 0.282·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.484·3-s + 0.5·4-s + 1.64·5-s − 0.342·6-s − 0.106·7-s − 0.353·8-s − 0.765·9-s − 1.16·10-s − 0.150·11-s + 0.242·12-s − 1.78·13-s + 0.0756·14-s + 0.796·15-s + 0.250·16-s − 1.07·17-s + 0.541·18-s + 1.41·19-s + 0.822·20-s − 0.0518·21-s + 0.106·22-s − 0.208·23-s − 0.171·24-s + 1.70·25-s + 1.26·26-s − 0.855·27-s − 0.0534·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.935426558\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.935426558\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 131 | \( 1 - T \) |
good | 3 | \( 1 - 0.839T + 3T^{2} \) |
| 5 | \( 1 - 3.67T + 5T^{2} \) |
| 7 | \( 1 + 0.282T + 7T^{2} \) |
| 11 | \( 1 + 0.500T + 11T^{2} \) |
| 13 | \( 1 + 6.43T + 13T^{2} \) |
| 17 | \( 1 + 4.42T + 17T^{2} \) |
| 19 | \( 1 - 6.17T + 19T^{2} \) |
| 29 | \( 1 - 2.78T + 29T^{2} \) |
| 31 | \( 1 - 6.42T + 31T^{2} \) |
| 37 | \( 1 - 10.5T + 37T^{2} \) |
| 41 | \( 1 - 8.71T + 41T^{2} \) |
| 43 | \( 1 + 4.30T + 43T^{2} \) |
| 47 | \( 1 + 1.92T + 47T^{2} \) |
| 53 | \( 1 - 4.60T + 53T^{2} \) |
| 59 | \( 1 + 11.7T + 59T^{2} \) |
| 61 | \( 1 - 5.46T + 61T^{2} \) |
| 67 | \( 1 + 9.53T + 67T^{2} \) |
| 71 | \( 1 - 10.6T + 71T^{2} \) |
| 73 | \( 1 - 13.5T + 73T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 - 14.8T + 83T^{2} \) |
| 89 | \( 1 - 6.41T + 89T^{2} \) |
| 97 | \( 1 + 4.54T + 97T^{2} \) |
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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
−8.002861112281739983301855655439, −7.61668065828204846453514517450, −6.55124211892876508928796545260, −6.17494296994506604231118891610, −5.24844149345406986667518736732, −4.70874587504892492686133849681, −3.15816523203554316576063450382, −2.47118457376042476093199572946, −2.12110160113300943883174697034, −0.76684536799552810711487117372,
0.76684536799552810711487117372, 2.12110160113300943883174697034, 2.47118457376042476093199572946, 3.15816523203554316576063450382, 4.70874587504892492686133849681, 5.24844149345406986667518736732, 6.17494296994506604231118891610, 6.55124211892876508928796545260, 7.61668065828204846453514517450, 8.002861112281739983301855655439