| L(s) = 1 | − 2.34·2-s + 3.48·4-s + 0.146·5-s + 7-s − 3.48·8-s − 0.342·10-s + 4.34·13-s − 2.34·14-s + 1.19·16-s + 0.146·17-s − 1.83·19-s + 0.510·20-s − 8.81·23-s − 4.97·25-s − 10.1·26-s + 3.48·28-s − 4.34·29-s + 0.292·31-s + 4.17·32-s − 0.342·34-s + 0.146·35-s − 3.48·37-s + 4.29·38-s − 0.510·40-s − 2.80·41-s − 7.86·43-s + 20.6·46-s + ⋯ |
| L(s) = 1 | − 1.65·2-s + 1.74·4-s + 0.0654·5-s + 0.377·7-s − 1.23·8-s − 0.108·10-s + 1.20·13-s − 0.626·14-s + 0.299·16-s + 0.0354·17-s − 0.420·19-s + 0.114·20-s − 1.83·23-s − 0.995·25-s − 1.99·26-s + 0.659·28-s − 0.806·29-s + 0.0525·31-s + 0.738·32-s − 0.0588·34-s + 0.0247·35-s − 0.573·37-s + 0.696·38-s − 0.0807·40-s − 0.437·41-s − 1.19·43-s + 3.04·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7191026138\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7191026138\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 2.34T + 2T^{2} \) |
| 5 | \( 1 - 0.146T + 5T^{2} \) |
| 13 | \( 1 - 4.34T + 13T^{2} \) |
| 17 | \( 1 - 0.146T + 17T^{2} \) |
| 19 | \( 1 + 1.83T + 19T^{2} \) |
| 23 | \( 1 + 8.81T + 23T^{2} \) |
| 29 | \( 1 + 4.34T + 29T^{2} \) |
| 31 | \( 1 - 0.292T + 31T^{2} \) |
| 37 | \( 1 + 3.48T + 37T^{2} \) |
| 41 | \( 1 + 2.80T + 41T^{2} \) |
| 43 | \( 1 + 7.86T + 43T^{2} \) |
| 47 | \( 1 - 0.949T + 47T^{2} \) |
| 53 | \( 1 - 4.51T + 53T^{2} \) |
| 59 | \( 1 + 8.02T + 59T^{2} \) |
| 61 | \( 1 - 5.43T + 61T^{2} \) |
| 67 | \( 1 + 7.76T + 67T^{2} \) |
| 71 | \( 1 - 3.53T + 71T^{2} \) |
| 73 | \( 1 - 16.1T + 73T^{2} \) |
| 79 | \( 1 - 13.0T + 79T^{2} \) |
| 83 | \( 1 - 12.9T + 83T^{2} \) |
| 89 | \( 1 - 9.81T + 89T^{2} \) |
| 97 | \( 1 + 17.4T + 97T^{2} \) |
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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
−8.049082626568315677663102657201, −7.54419222383264713413522037491, −6.56934948232797859229671098707, −6.15395162640469822584308006228, −5.26496533129620963729649557169, −4.13620062316898863243895961219, −3.44571361077046265682679334400, −2.06396636976327026727194185873, −1.75194926047249391797269885618, −0.54435252349942297671831475387,
0.54435252349942297671831475387, 1.75194926047249391797269885618, 2.06396636976327026727194185873, 3.44571361077046265682679334400, 4.13620062316898863243895961219, 5.26496533129620963729649557169, 6.15395162640469822584308006228, 6.56934948232797859229671098707, 7.54419222383264713413522037491, 8.049082626568315677663102657201
