| L(s) = 1 | − 2.04·2-s + 2.17·4-s − 1.43·7-s − 0.364·8-s − 5.70·11-s + 2·13-s + 2.93·14-s − 3.61·16-s + 3.35·17-s + 19-s + 11.6·22-s − 9.06·23-s − 4.08·26-s − 3.12·28-s − 4.45·29-s − 4.79·31-s + 8.11·32-s − 6.86·34-s + 4.35·37-s − 2.04·38-s + 8.69·41-s + 8.86·43-s − 12.4·44-s + 18.5·46-s − 3.22·47-s − 4.94·49-s + 4.35·52-s + ⋯ |
| L(s) = 1 | − 1.44·2-s + 1.08·4-s − 0.541·7-s − 0.128·8-s − 1.71·11-s + 0.554·13-s + 0.783·14-s − 0.902·16-s + 0.814·17-s + 0.229·19-s + 2.48·22-s − 1.88·23-s − 0.801·26-s − 0.590·28-s − 0.826·29-s − 0.860·31-s + 1.43·32-s − 1.17·34-s + 0.716·37-s − 0.331·38-s + 1.35·41-s + 1.35·43-s − 1.87·44-s + 2.73·46-s − 0.471·47-s − 0.706·49-s + 0.604·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4449812994\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4449812994\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + 2.04T + 2T^{2} \) |
| 7 | \( 1 + 1.43T + 7T^{2} \) |
| 11 | \( 1 + 5.70T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 3.35T + 17T^{2} \) |
| 23 | \( 1 + 9.06T + 23T^{2} \) |
| 29 | \( 1 + 4.45T + 29T^{2} \) |
| 31 | \( 1 + 4.79T + 31T^{2} \) |
| 37 | \( 1 - 4.35T + 37T^{2} \) |
| 41 | \( 1 - 8.69T + 41T^{2} \) |
| 43 | \( 1 - 8.86T + 43T^{2} \) |
| 47 | \( 1 + 3.22T + 47T^{2} \) |
| 53 | \( 1 + 3.72T + 53T^{2} \) |
| 59 | \( 1 + 12.7T + 59T^{2} \) |
| 61 | \( 1 + 15.4T + 61T^{2} \) |
| 67 | \( 1 - 13.6T + 67T^{2} \) |
| 71 | \( 1 + 0.652T + 71T^{2} \) |
| 73 | \( 1 - 9.86T + 73T^{2} \) |
| 79 | \( 1 - 9.14T + 79T^{2} \) |
| 83 | \( 1 + 12.4T + 83T^{2} \) |
| 89 | \( 1 + 3.72T + 89T^{2} \) |
| 97 | \( 1 + 6.09T + 97T^{2} \) |
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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
−8.261561420630998798893077954232, −7.72206820317365686114216310465, −7.48411089862524107059840972643, −6.21345852659358065598963399361, −5.74117474572460121407583478292, −4.67417301869428646831655919657, −3.61447981884561330123181831637, −2.62800644816284939204749460101, −1.74886687956299000441981357653, −0.45874616880745029294679836334,
0.45874616880745029294679836334, 1.74886687956299000441981357653, 2.62800644816284939204749460101, 3.61447981884561330123181831637, 4.67417301869428646831655919657, 5.74117474572460121407583478292, 6.21345852659358065598963399361, 7.48411089862524107059840972643, 7.72206820317365686114216310465, 8.261561420630998798893077954232
