| L(s) = 1 | − 2.35·2-s + 3.54·4-s − 0.0127·7-s − 3.63·8-s − 0.615·11-s + 3.05·13-s + 0.0301·14-s + 1.46·16-s + 6.63·17-s − 3.54·19-s + 1.44·22-s + 1.27·23-s − 7.19·26-s − 0.0453·28-s + 29-s + 6.68·31-s + 3.81·32-s − 15.6·34-s + 6.22·37-s + 8.33·38-s − 7.00·41-s + 5.31·43-s − 2.18·44-s − 3.00·46-s − 3.22·47-s − 6.99·49-s + 10.8·52-s + ⋯ |
| L(s) = 1 | − 1.66·2-s + 1.77·4-s − 0.00483·7-s − 1.28·8-s − 0.185·11-s + 0.847·13-s + 0.00805·14-s + 0.365·16-s + 1.60·17-s − 0.812·19-s + 0.309·22-s + 0.266·23-s − 1.41·26-s − 0.00856·28-s + 0.185·29-s + 1.20·31-s + 0.674·32-s − 2.67·34-s + 1.02·37-s + 1.35·38-s − 1.09·41-s + 0.809·43-s − 0.328·44-s − 0.442·46-s − 0.469·47-s − 0.999·49-s + 1.50·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9412456308\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9412456308\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + 2.35T + 2T^{2} \) |
| 7 | \( 1 + 0.0127T + 7T^{2} \) |
| 11 | \( 1 + 0.615T + 11T^{2} \) |
| 13 | \( 1 - 3.05T + 13T^{2} \) |
| 17 | \( 1 - 6.63T + 17T^{2} \) |
| 19 | \( 1 + 3.54T + 19T^{2} \) |
| 23 | \( 1 - 1.27T + 23T^{2} \) |
| 31 | \( 1 - 6.68T + 31T^{2} \) |
| 37 | \( 1 - 6.22T + 37T^{2} \) |
| 41 | \( 1 + 7.00T + 41T^{2} \) |
| 43 | \( 1 - 5.31T + 43T^{2} \) |
| 47 | \( 1 + 3.22T + 47T^{2} \) |
| 53 | \( 1 - 3.77T + 53T^{2} \) |
| 59 | \( 1 - 8.04T + 59T^{2} \) |
| 61 | \( 1 + 0.250T + 61T^{2} \) |
| 67 | \( 1 - 2.90T + 67T^{2} \) |
| 71 | \( 1 - 14.0T + 71T^{2} \) |
| 73 | \( 1 - 8.75T + 73T^{2} \) |
| 79 | \( 1 + 8.34T + 79T^{2} \) |
| 83 | \( 1 + 16.2T + 83T^{2} \) |
| 89 | \( 1 + 15.2T + 89T^{2} \) |
| 97 | \( 1 + 9.90T + 97T^{2} \) |
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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
−8.300913814670430257468156198250, −7.56985124342910648144388324113, −6.78467118276758417430871435853, −6.20814793780265496058745837943, −5.37984147013073013924446579446, −4.33514444905714353388357482222, −3.32268941657840700235809532326, −2.47285968606597207552342776126, −1.44602056736689249384627155893, −0.70340844892407912700692815584,
0.70340844892407912700692815584, 1.44602056736689249384627155893, 2.47285968606597207552342776126, 3.32268941657840700235809532326, 4.33514444905714353388357482222, 5.37984147013073013924446579446, 6.20814793780265496058745837943, 6.78467118276758417430871435853, 7.56985124342910648144388324113, 8.300913814670430257468156198250
