| L(s) = 1 | + 4·2-s + 3·3-s + 16·4-s + 12·6-s − 49·7-s + 64·8-s − 234·9-s + 405·11-s + 48·12-s + 391·13-s − 196·14-s + 256·16-s − 999·17-s − 936·18-s + 2.34e3·19-s − 147·21-s + 1.62e3·22-s − 2.43e3·23-s + 192·24-s + 1.56e3·26-s − 1.43e3·27-s − 784·28-s + 8.25e3·29-s + 4.01e3·31-s + 1.02e3·32-s + 1.21e3·33-s − 3.99e3·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.192·3-s + 1/2·4-s + 0.136·6-s − 0.377·7-s + 0.353·8-s − 0.962·9-s + 1.00·11-s + 0.0962·12-s + 0.641·13-s − 0.267·14-s + 1/4·16-s − 0.838·17-s − 0.680·18-s + 1.48·19-s − 0.0727·21-s + 0.713·22-s − 0.957·23-s + 0.0680·24-s + 0.453·26-s − 0.377·27-s − 0.188·28-s + 1.82·29-s + 0.750·31-s + 0.176·32-s + 0.194·33-s − 0.592·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.580355452\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.580355452\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{2} T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + p^{2} T \) |
| good | 3 | \( 1 - p T + p^{5} T^{2} \) |
| 11 | \( 1 - 405 T + p^{5} T^{2} \) |
| 13 | \( 1 - 391 T + p^{5} T^{2} \) |
| 17 | \( 1 + 999 T + p^{5} T^{2} \) |
| 19 | \( 1 - 2342 T + p^{5} T^{2} \) |
| 23 | \( 1 + 2430 T + p^{5} T^{2} \) |
| 29 | \( 1 - 8259 T + p^{5} T^{2} \) |
| 31 | \( 1 - 4016 T + p^{5} T^{2} \) |
| 37 | \( 1 - 7042 T + p^{5} T^{2} \) |
| 41 | \( 1 - 3336 T + p^{5} T^{2} \) |
| 43 | \( 1 - 23518 T + p^{5} T^{2} \) |
| 47 | \( 1 + 10317 T + p^{5} T^{2} \) |
| 53 | \( 1 + 3084 T + p^{5} T^{2} \) |
| 59 | \( 1 + 18816 T + p^{5} T^{2} \) |
| 61 | \( 1 - 21668 T + p^{5} T^{2} \) |
| 67 | \( 1 + 52124 T + p^{5} T^{2} \) |
| 71 | \( 1 + 28560 T + p^{5} T^{2} \) |
| 73 | \( 1 - 70342 T + p^{5} T^{2} \) |
| 79 | \( 1 - 58823 T + p^{5} T^{2} \) |
| 83 | \( 1 + 756 T + p^{5} T^{2} \) |
| 89 | \( 1 - 135384 T + p^{5} T^{2} \) |
| 97 | \( 1 + 110435 T + p^{5} T^{2} \) |
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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
−10.90887019790715533288427088135, −9.694422397416912170439524183687, −8.811410160009475720786102582682, −7.79074303602366197939087617792, −6.49866103776842348936513076150, −5.93064066590837774952297739197, −4.58629880375921290546878883187, −3.51968372649651468839777534565, −2.53559006144540271591301503518, −0.947634133290767031152071149240,
0.947634133290767031152071149240, 2.53559006144540271591301503518, 3.51968372649651468839777534565, 4.58629880375921290546878883187, 5.93064066590837774952297739197, 6.49866103776842348936513076150, 7.79074303602366197939087617792, 8.811410160009475720786102582682, 9.694422397416912170439524183687, 10.90887019790715533288427088135
