| L(s) = 1 | + 2.61·2-s + 4.85·4-s + 2.23·5-s + 7.47·8-s + 5.85·10-s + 11-s + 3.47·13-s + 9.85·16-s − 6·17-s − 4.23·19-s + 10.8·20-s + 2.61·22-s + 2.47·23-s + 9.09·26-s + 10.2·29-s + 8.47·31-s + 10.8·32-s − 15.7·34-s − 0.527·37-s − 11.0·38-s + 16.7·40-s − 6·41-s − 0.472·43-s + 4.85·44-s + 6.47·46-s − 11.4·47-s + 16.8·52-s + ⋯ |
| L(s) = 1 | + 1.85·2-s + 2.42·4-s + 0.999·5-s + 2.64·8-s + 1.85·10-s + 0.301·11-s + 0.962·13-s + 2.46·16-s − 1.45·17-s − 0.971·19-s + 2.42·20-s + 0.558·22-s + 0.515·23-s + 1.78·26-s + 1.90·29-s + 1.52·31-s + 1.91·32-s − 2.69·34-s − 0.0867·37-s − 1.79·38-s + 2.64·40-s − 0.937·41-s − 0.0720·43-s + 0.731·44-s + 0.954·46-s − 1.67·47-s + 2.33·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4851 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4851 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.348271430\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.348271430\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 - 2.61T + 2T^{2} \) |
| 5 | \( 1 - 2.23T + 5T^{2} \) |
| 13 | \( 1 - 3.47T + 13T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 + 4.23T + 19T^{2} \) |
| 23 | \( 1 - 2.47T + 23T^{2} \) |
| 29 | \( 1 - 10.2T + 29T^{2} \) |
| 31 | \( 1 - 8.47T + 31T^{2} \) |
| 37 | \( 1 + 0.527T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 0.472T + 43T^{2} \) |
| 47 | \( 1 + 11.4T + 47T^{2} \) |
| 53 | \( 1 - 12.9T + 53T^{2} \) |
| 59 | \( 1 - 11.9T + 59T^{2} \) |
| 61 | \( 1 + 6.94T + 61T^{2} \) |
| 67 | \( 1 - 0.708T + 67T^{2} \) |
| 71 | \( 1 + 4.47T + 71T^{2} \) |
| 73 | \( 1 + T + 73T^{2} \) |
| 79 | \( 1 + 2.47T + 79T^{2} \) |
| 83 | \( 1 + 12.4T + 83T^{2} \) |
| 89 | \( 1 - 13.4T + 89T^{2} \) |
| 97 | \( 1 + 6.94T + 97T^{2} \) |
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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
−8.310433093170564276830090411001, −6.91439035429333203162134790866, −6.51114915634020253482846225053, −6.13706364626533386518532266294, −5.24235802543203367006570695931, −4.55843527103297888703111719756, −3.97262724535007170213709378782, −2.92705263726679144138433776591, −2.29174517526254193813568760293, −1.38579357828222635179628027076,
1.38579357828222635179628027076, 2.29174517526254193813568760293, 2.92705263726679144138433776591, 3.97262724535007170213709378782, 4.55843527103297888703111719756, 5.24235802543203367006570695931, 6.13706364626533386518532266294, 6.51114915634020253482846225053, 6.91439035429333203162134790866, 8.310433093170564276830090411001
