| L(s) = 1 | + 3·3-s + 3·5-s − 2·7-s + 6·9-s + 11-s − 3·13-s + 9·15-s − 4·17-s + 8·19-s − 6·21-s + 4·25-s + 9·27-s + 29-s + 3·31-s + 3·33-s − 6·35-s + 8·37-s − 9·39-s − 2·41-s − 7·43-s + 18·45-s + 11·47-s − 3·49-s − 12·51-s − 53-s + 3·55-s + 24·57-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 1.34·5-s − 0.755·7-s + 2·9-s + 0.301·11-s − 0.832·13-s + 2.32·15-s − 0.970·17-s + 1.83·19-s − 1.30·21-s + 4/5·25-s + 1.73·27-s + 0.185·29-s + 0.538·31-s + 0.522·33-s − 1.01·35-s + 1.31·37-s − 1.44·39-s − 0.312·41-s − 1.06·43-s + 2.68·45-s + 1.60·47-s − 3/7·49-s − 1.68·51-s − 0.137·53-s + 0.404·55-s + 3.17·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.864756953\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.864756953\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 - 11 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 + 7 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−9.282938737818127575750781224601, −8.719436578087282195198988705982, −7.69215062619124894804705063311, −7.01001452094158583370492958723, −6.19575721253383792338242213991, −5.13333342499651093614202490086, −4.06838015586268796258310338440, −2.95884490136083113636934725479, −2.51885917774288386423293648527, −1.44035765383180978016313632525,
1.44035765383180978016313632525, 2.51885917774288386423293648527, 2.95884490136083113636934725479, 4.06838015586268796258310338440, 5.13333342499651093614202490086, 6.19575721253383792338242213991, 7.01001452094158583370492958723, 7.69215062619124894804705063311, 8.719436578087282195198988705982, 9.282938737818127575750781224601
