| L(s) = 1 | − 2.69·2-s + 5.27·4-s + 3.56·7-s − 8.84·8-s + 0.0695·11-s − 4.85·13-s − 9.62·14-s + 13.2·16-s + 3.03·17-s + 5.05·19-s − 0.187·22-s − 7.43·23-s + 13.1·26-s + 18.8·28-s − 1.95·29-s − 5.63·31-s − 18.1·32-s − 8.18·34-s − 6.21·37-s − 13.6·38-s + 5.63·41-s − 0.244·43-s + 0.367·44-s + 20.0·46-s − 3.23·47-s + 5.71·49-s − 25.6·52-s + ⋯ |
| L(s) = 1 | − 1.90·2-s + 2.63·4-s + 1.34·7-s − 3.12·8-s + 0.0209·11-s − 1.34·13-s − 2.57·14-s + 3.32·16-s + 0.735·17-s + 1.15·19-s − 0.0400·22-s − 1.54·23-s + 2.56·26-s + 3.55·28-s − 0.362·29-s − 1.01·31-s − 3.21·32-s − 1.40·34-s − 1.02·37-s − 2.21·38-s + 0.880·41-s − 0.0372·43-s + 0.0553·44-s + 2.95·46-s − 0.471·47-s + 0.817·49-s − 3.55·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 2.69T + 2T^{2} \) |
| 7 | \( 1 - 3.56T + 7T^{2} \) |
| 11 | \( 1 - 0.0695T + 11T^{2} \) |
| 13 | \( 1 + 4.85T + 13T^{2} \) |
| 17 | \( 1 - 3.03T + 17T^{2} \) |
| 19 | \( 1 - 5.05T + 19T^{2} \) |
| 23 | \( 1 + 7.43T + 23T^{2} \) |
| 29 | \( 1 + 1.95T + 29T^{2} \) |
| 31 | \( 1 + 5.63T + 31T^{2} \) |
| 37 | \( 1 + 6.21T + 37T^{2} \) |
| 41 | \( 1 - 5.63T + 41T^{2} \) |
| 43 | \( 1 + 0.244T + 43T^{2} \) |
| 47 | \( 1 + 3.23T + 47T^{2} \) |
| 53 | \( 1 - 8.37T + 53T^{2} \) |
| 59 | \( 1 - 1.60T + 59T^{2} \) |
| 61 | \( 1 - 3.12T + 61T^{2} \) |
| 67 | \( 1 - 2.94T + 67T^{2} \) |
| 71 | \( 1 + 7.25T + 71T^{2} \) |
| 73 | \( 1 + 3.69T + 73T^{2} \) |
| 79 | \( 1 - 4.70T + 79T^{2} \) |
| 83 | \( 1 + 8.80T + 83T^{2} \) |
| 89 | \( 1 - 3.55T + 89T^{2} \) |
| 97 | \( 1 + 8.80T + 97T^{2} \) |
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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
−7.73861047825245432169744337462, −7.54522789117889521416359696333, −6.81755791803464452272367800660, −5.67716493700590179602302357165, −5.20738220539036120779345645516, −3.91912796230620818832224650891, −2.74096162697275333777004906385, −1.95633770104611575952311389096, −1.25967463027440708238426845823, 0,
1.25967463027440708238426845823, 1.95633770104611575952311389096, 2.74096162697275333777004906385, 3.91912796230620818832224650891, 5.20738220539036120779345645516, 5.67716493700590179602302357165, 6.81755791803464452272367800660, 7.54522789117889521416359696333, 7.73861047825245432169744337462
