| L(s) = 1 | + 2·2-s + 7·3-s + 4·4-s + 14·6-s − 34·7-s + 8·8-s + 22·9-s + 27·11-s + 28·12-s − 28·13-s − 68·14-s + 16·16-s + 21·17-s + 44·18-s + 35·19-s − 238·21-s + 54·22-s − 78·23-s + 56·24-s − 56·26-s − 35·27-s − 136·28-s − 120·29-s + 182·31-s + 32·32-s + 189·33-s + 42·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.34·3-s + 1/2·4-s + 0.952·6-s − 1.83·7-s + 0.353·8-s + 0.814·9-s + 0.740·11-s + 0.673·12-s − 0.597·13-s − 1.29·14-s + 1/4·16-s + 0.299·17-s + 0.576·18-s + 0.422·19-s − 2.47·21-s + 0.523·22-s − 0.707·23-s + 0.476·24-s − 0.422·26-s − 0.249·27-s − 0.917·28-s − 0.768·29-s + 1.05·31-s + 0.176·32-s + 0.996·33-s + 0.211·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.492197210\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.492197210\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 7 T + p^{3} T^{2} \) |
| 7 | \( 1 + 34 T + p^{3} T^{2} \) |
| 11 | \( 1 - 27 T + p^{3} T^{2} \) |
| 13 | \( 1 + 28 T + p^{3} T^{2} \) |
| 17 | \( 1 - 21 T + p^{3} T^{2} \) |
| 19 | \( 1 - 35 T + p^{3} T^{2} \) |
| 23 | \( 1 + 78 T + p^{3} T^{2} \) |
| 29 | \( 1 + 120 T + p^{3} T^{2} \) |
| 31 | \( 1 - 182 T + p^{3} T^{2} \) |
| 37 | \( 1 - 146 T + p^{3} T^{2} \) |
| 41 | \( 1 - 357 T + p^{3} T^{2} \) |
| 43 | \( 1 + 148 T + p^{3} T^{2} \) |
| 47 | \( 1 + 84 T + p^{3} T^{2} \) |
| 53 | \( 1 - 702 T + p^{3} T^{2} \) |
| 59 | \( 1 + 840 T + p^{3} T^{2} \) |
| 61 | \( 1 + 238 T + p^{3} T^{2} \) |
| 67 | \( 1 - 461 T + p^{3} T^{2} \) |
| 71 | \( 1 + 708 T + p^{3} T^{2} \) |
| 73 | \( 1 + 133 T + p^{3} T^{2} \) |
| 79 | \( 1 - 650 T + p^{3} T^{2} \) |
| 83 | \( 1 + 903 T + p^{3} T^{2} \) |
| 89 | \( 1 - 735 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1106 T + p^{3} T^{2} \) |
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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
−14.85968603147664231141558773019, −13.89404450416413533969944784393, −13.06059084228858212332733008637, −12.00590856826000061959897808646, −9.987039499648964655581028278805, −9.161232348529186663694273724453, −7.50540663806772273306522443384, −6.17722218497597972994388809641, −3.84905882882645491861869169033, −2.75166401889686716685696480861,
2.75166401889686716685696480861, 3.84905882882645491861869169033, 6.17722218497597972994388809641, 7.50540663806772273306522443384, 9.161232348529186663694273724453, 9.987039499648964655581028278805, 12.00590856826000061959897808646, 13.06059084228858212332733008637, 13.89404450416413533969944784393, 14.85968603147664231141558773019
