| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 0.381·7-s − 8-s + 9-s + 1.38·11-s + 12-s − 2.47·13-s + 0.381·14-s + 16-s + 3.23·17-s − 18-s + 7.70·19-s − 0.381·21-s − 1.38·22-s + 4.47·23-s − 24-s + 2.47·26-s + 27-s − 0.381·28-s − 0.472·29-s + 4.38·31-s − 32-s + 1.38·33-s − 3.23·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.408·6-s − 0.144·7-s − 0.353·8-s + 0.333·9-s + 0.416·11-s + 0.288·12-s − 0.685·13-s + 0.102·14-s + 0.250·16-s + 0.784·17-s − 0.235·18-s + 1.76·19-s − 0.0833·21-s − 0.294·22-s + 0.932·23-s − 0.204·24-s + 0.484·26-s + 0.192·27-s − 0.0721·28-s − 0.0876·29-s + 0.787·31-s − 0.176·32-s + 0.240·33-s − 0.554·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.796241218\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.796241218\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 0.381T + 7T^{2} \) |
| 11 | \( 1 - 1.38T + 11T^{2} \) |
| 13 | \( 1 + 2.47T + 13T^{2} \) |
| 17 | \( 1 - 3.23T + 17T^{2} \) |
| 19 | \( 1 - 7.70T + 19T^{2} \) |
| 23 | \( 1 - 4.47T + 23T^{2} \) |
| 29 | \( 1 + 0.472T + 29T^{2} \) |
| 31 | \( 1 - 4.38T + 31T^{2} \) |
| 37 | \( 1 + 8T + 37T^{2} \) |
| 41 | \( 1 + 7.70T + 41T^{2} \) |
| 43 | \( 1 + 5.70T + 43T^{2} \) |
| 47 | \( 1 - 11.7T + 47T^{2} \) |
| 53 | \( 1 + 9.09T + 53T^{2} \) |
| 59 | \( 1 + 1.38T + 59T^{2} \) |
| 61 | \( 1 - 7.23T + 61T^{2} \) |
| 67 | \( 1 - 10.4T + 67T^{2} \) |
| 71 | \( 1 - 14.4T + 71T^{2} \) |
| 73 | \( 1 + 12.4T + 73T^{2} \) |
| 79 | \( 1 + 3.38T + 79T^{2} \) |
| 83 | \( 1 + 8.85T + 83T^{2} \) |
| 89 | \( 1 - 12.4T + 89T^{2} \) |
| 97 | \( 1 - 5.61T + 97T^{2} \) |
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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
−8.570642040566707803046287899006, −7.80179931616454416338550187124, −7.20199184242200679702685729682, −6.61754745790366784493018898209, −5.48640696120919602949863298103, −4.82561464020740528134891758908, −3.47929116784812959822135759169, −3.04596585279728511073331984621, −1.86876960589175213100467772567, −0.873034126105967997488391100681,
0.873034126105967997488391100681, 1.86876960589175213100467772567, 3.04596585279728511073331984621, 3.47929116784812959822135759169, 4.82561464020740528134891758908, 5.48640696120919602949863298103, 6.61754745790366784493018898209, 7.20199184242200679702685729682, 7.80179931616454416338550187124, 8.570642040566707803046287899006
