| L(s) = 1 | + 2·2-s + 5·3-s + 4·4-s + 9·5-s + 10·6-s + 8·8-s − 2·9-s + 18·10-s − 57·11-s + 20·12-s + 70·13-s + 45·15-s + 16·16-s − 51·17-s − 4·18-s − 5·19-s + 36·20-s − 114·22-s + 69·23-s + 40·24-s − 44·25-s + 140·26-s − 145·27-s + 114·29-s + 90·30-s − 23·31-s + 32·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.962·3-s + 1/2·4-s + 0.804·5-s + 0.680·6-s + 0.353·8-s − 0.0740·9-s + 0.569·10-s − 1.56·11-s + 0.481·12-s + 1.49·13-s + 0.774·15-s + 1/4·16-s − 0.727·17-s − 0.0523·18-s − 0.0603·19-s + 0.402·20-s − 1.10·22-s + 0.625·23-s + 0.340·24-s − 0.351·25-s + 1.05·26-s − 1.03·27-s + 0.729·29-s + 0.547·30-s − 0.133·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.185692354\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.185692354\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 5 T + p^{3} T^{2} \) |
| 5 | \( 1 - 9 T + p^{3} T^{2} \) |
| 11 | \( 1 + 57 T + p^{3} T^{2} \) |
| 13 | \( 1 - 70 T + p^{3} T^{2} \) |
| 17 | \( 1 + 3 p T + p^{3} T^{2} \) |
| 19 | \( 1 + 5 T + p^{3} T^{2} \) |
| 23 | \( 1 - 3 p T + p^{3} T^{2} \) |
| 29 | \( 1 - 114 T + p^{3} T^{2} \) |
| 31 | \( 1 + 23 T + p^{3} T^{2} \) |
| 37 | \( 1 + 253 T + p^{3} T^{2} \) |
| 41 | \( 1 - 42 T + p^{3} T^{2} \) |
| 43 | \( 1 + 124 T + p^{3} T^{2} \) |
| 47 | \( 1 + 201 T + p^{3} T^{2} \) |
| 53 | \( 1 + 393 T + p^{3} T^{2} \) |
| 59 | \( 1 + 219 T + p^{3} T^{2} \) |
| 61 | \( 1 - 709 T + p^{3} T^{2} \) |
| 67 | \( 1 - 419 T + p^{3} T^{2} \) |
| 71 | \( 1 + 96 T + p^{3} T^{2} \) |
| 73 | \( 1 - 313 T + p^{3} T^{2} \) |
| 79 | \( 1 - 461 T + p^{3} T^{2} \) |
| 83 | \( 1 - 588 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1017 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1834 T + p^{3} T^{2} \) |
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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
−13.47478173650198898543119590834, −12.95907436692134841053560085563, −11.26675438061777843734757808862, −10.29174754405156514345866546626, −8.909623836493129563407829098705, −7.952907508488061201484123215963, −6.35997490773533338162821223751, −5.17201496260031785952011677280, −3.38653008273238754791285314334, −2.16482003301400300720608945723,
2.16482003301400300720608945723, 3.38653008273238754791285314334, 5.17201496260031785952011677280, 6.35997490773533338162821223751, 7.952907508488061201484123215963, 8.909623836493129563407829098705, 10.29174754405156514345866546626, 11.26675438061777843734757808862, 12.95907436692134841053560085563, 13.47478173650198898543119590834
