| L(s) = 1 | + 2.37·2-s + 1.27·3-s + 3.65·4-s + 3.02·6-s + 0.726·7-s + 3.92·8-s − 1.37·9-s − 0.273·11-s + 4.65·12-s − 5.95·13-s + 1.72·14-s + 2.02·16-s + 5.27·17-s − 3.27·18-s + 19-s + 0.924·21-s − 0.651·22-s − 3.67·23-s + 4.99·24-s − 14.1·26-s − 5.57·27-s + 2.65·28-s − 2.27·29-s + 3.19·31-s − 3.02·32-s − 0.348·33-s + 12.5·34-s + ⋯ |
| L(s) = 1 | + 1.68·2-s + 0.735·3-s + 1.82·4-s + 1.23·6-s + 0.274·7-s + 1.38·8-s − 0.459·9-s − 0.0825·11-s + 1.34·12-s − 1.65·13-s + 0.461·14-s + 0.507·16-s + 1.27·17-s − 0.771·18-s + 0.229·19-s + 0.201·21-s − 0.138·22-s − 0.767·23-s + 1.02·24-s − 2.77·26-s − 1.07·27-s + 0.501·28-s − 0.422·29-s + 0.574·31-s − 0.535·32-s − 0.0607·33-s + 2.15·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.059924695\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.059924695\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 2.37T + 2T^{2} \) |
| 3 | \( 1 - 1.27T + 3T^{2} \) |
| 7 | \( 1 - 0.726T + 7T^{2} \) |
| 11 | \( 1 + 0.273T + 11T^{2} \) |
| 13 | \( 1 + 5.95T + 13T^{2} \) |
| 17 | \( 1 - 5.27T + 17T^{2} \) |
| 23 | \( 1 + 3.67T + 23T^{2} \) |
| 29 | \( 1 + 2.27T + 29T^{2} \) |
| 31 | \( 1 - 3.19T + 31T^{2} \) |
| 37 | \( 1 - 8.12T + 37T^{2} \) |
| 41 | \( 1 + 9.43T + 41T^{2} \) |
| 43 | \( 1 - 9.81T + 43T^{2} \) |
| 47 | \( 1 - 12.1T + 47T^{2} \) |
| 53 | \( 1 - 5.69T + 53T^{2} \) |
| 59 | \( 1 + 4.20T + 59T^{2} \) |
| 61 | \( 1 + 0.103T + 61T^{2} \) |
| 67 | \( 1 + 11.7T + 67T^{2} \) |
| 71 | \( 1 - 5.75T + 71T^{2} \) |
| 73 | \( 1 - 6.67T + 73T^{2} \) |
| 79 | \( 1 - 3.87T + 79T^{2} \) |
| 83 | \( 1 + 0.488T + 83T^{2} \) |
| 89 | \( 1 + 16.4T + 89T^{2} \) |
| 97 | \( 1 + 4.44T + 97T^{2} \) |
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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
−11.46565112135528399482915620073, −10.22832246189667208581791355391, −9.287547370050828083417562098063, −7.948930766172280682723796366646, −7.31288573754649677519000114720, −5.96676809676042291135130153753, −5.21816996360135943270646722740, −4.19200919580490190485706042531, −3.08706149699183114976329639808, −2.28433241393317875635177771980,
2.28433241393317875635177771980, 3.08706149699183114976329639808, 4.19200919580490190485706042531, 5.21816996360135943270646722740, 5.96676809676042291135130153753, 7.31288573754649677519000114720, 7.948930766172280682723796366646, 9.287547370050828083417562098063, 10.22832246189667208581791355391, 11.46565112135528399482915620073
