| L(s) = 1 | − 4.82·2-s − 3·3-s + 15.2·4-s + 14.4·6-s − 14.5·7-s − 35.1·8-s + 9·9-s + 10.0·11-s − 45.8·12-s − 5.07·13-s + 70.4·14-s + 47.2·16-s − 97.5·17-s − 43.4·18-s − 58.7·19-s + 43.7·21-s − 48.6·22-s − 15.8·23-s + 105.·24-s + 24.4·26-s − 27·27-s − 223.·28-s − 220.·29-s + 196.·31-s + 53.1·32-s − 30.2·33-s + 470.·34-s + ⋯ |
| L(s) = 1 | − 1.70·2-s − 0.577·3-s + 1.90·4-s + 0.984·6-s − 0.788·7-s − 1.55·8-s + 0.333·9-s + 0.276·11-s − 1.10·12-s − 0.108·13-s + 1.34·14-s + 0.737·16-s − 1.39·17-s − 0.568·18-s − 0.708·19-s + 0.455·21-s − 0.471·22-s − 0.143·23-s + 0.896·24-s + 0.184·26-s − 0.192·27-s − 1.50·28-s − 1.41·29-s + 1.14·31-s + 0.293·32-s − 0.159·33-s + 2.37·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1436806808\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1436806808\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 4.82T + 8T^{2} \) |
| 7 | \( 1 + 14.5T + 343T^{2} \) |
| 11 | \( 1 - 10.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 5.07T + 2.19e3T^{2} \) |
| 17 | \( 1 + 97.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 58.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 15.8T + 1.21e4T^{2} \) |
| 29 | \( 1 + 220.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 196.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 409.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 200.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 457.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 280.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 586.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 192.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 17.2T + 2.26e5T^{2} \) |
| 67 | \( 1 - 673.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 384.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 284.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 774.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.31e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 19.8T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.59e3T + 9.12e5T^{2} \) |
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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
−8.937174444535244860401458869718, −8.304902571165313601407277188703, −7.31547591400157120700816162478, −6.63949676362763063440779669102, −6.20454702028076569007748417857, −4.90857279907250833201500941415, −3.74834656635634967230596784733, −2.44161606488920839206529066324, −1.54344366899215347389351135076, −0.23619811479982645754101131914,
0.23619811479982645754101131914, 1.54344366899215347389351135076, 2.44161606488920839206529066324, 3.74834656635634967230596784733, 4.90857279907250833201500941415, 6.20454702028076569007748417857, 6.63949676362763063440779669102, 7.31547591400157120700816162478, 8.304902571165313601407277188703, 8.937174444535244860401458869718
