| L(s) = 1 | − 5·2-s + 7·3-s + 17·4-s + 7·5-s − 35·6-s − 45·8-s + 22·9-s − 35·10-s − 26·11-s + 119·12-s − 13·13-s + 49·15-s + 89·16-s − 77·17-s − 110·18-s + 126·19-s + 119·20-s + 130·22-s − 96·23-s − 315·24-s − 76·25-s + 65·26-s − 35·27-s − 82·29-s − 245·30-s − 196·31-s − 85·32-s + ⋯ |
| L(s) = 1 | − 1.76·2-s + 1.34·3-s + 17/8·4-s + 0.626·5-s − 2.38·6-s − 1.98·8-s + 0.814·9-s − 1.10·10-s − 0.712·11-s + 2.86·12-s − 0.277·13-s + 0.843·15-s + 1.39·16-s − 1.09·17-s − 1.44·18-s + 1.52·19-s + 1.33·20-s + 1.25·22-s − 0.870·23-s − 2.67·24-s − 0.607·25-s + 0.490·26-s − 0.249·27-s − 0.525·29-s − 1.49·30-s − 1.13·31-s − 0.469·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 13 | \( 1 + p T \) |
| good | 2 | \( 1 + 5 T + p^{3} T^{2} \) |
| 3 | \( 1 - 7 T + p^{3} T^{2} \) |
| 5 | \( 1 - 7 T + p^{3} T^{2} \) |
| 11 | \( 1 + 26 T + p^{3} T^{2} \) |
| 17 | \( 1 + 77 T + p^{3} T^{2} \) |
| 19 | \( 1 - 126 T + p^{3} T^{2} \) |
| 23 | \( 1 + 96 T + p^{3} T^{2} \) |
| 29 | \( 1 + 82 T + p^{3} T^{2} \) |
| 31 | \( 1 + 196 T + p^{3} T^{2} \) |
| 37 | \( 1 + 131 T + p^{3} T^{2} \) |
| 41 | \( 1 + 336 T + p^{3} T^{2} \) |
| 43 | \( 1 + 201 T + p^{3} T^{2} \) |
| 47 | \( 1 - 105 T + p^{3} T^{2} \) |
| 53 | \( 1 + 432 T + p^{3} T^{2} \) |
| 59 | \( 1 - 294 T + p^{3} T^{2} \) |
| 61 | \( 1 - 56 T + p^{3} T^{2} \) |
| 67 | \( 1 - 478 T + p^{3} T^{2} \) |
| 71 | \( 1 - 9 T + p^{3} T^{2} \) |
| 73 | \( 1 + 98 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1304 T + p^{3} T^{2} \) |
| 83 | \( 1 - 308 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1190 T + p^{3} T^{2} \) |
| 97 | \( 1 + 70 T + p^{3} T^{2} \) |
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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
−9.534893106935502474070915723996, −9.006334873726688441217695566847, −8.110606603563451634446516247844, −7.59900716652139842918036182426, −6.67178895170955588258204202108, −5.35739881565259158584368844178, −3.49108131744709760173443536467, −2.35687901654252125625048426133, −1.74396806429794314657777126362, 0,
1.74396806429794314657777126362, 2.35687901654252125625048426133, 3.49108131744709760173443536467, 5.35739881565259158584368844178, 6.67178895170955588258204202108, 7.59900716652139842918036182426, 8.110606603563451634446516247844, 9.006334873726688441217695566847, 9.534893106935502474070915723996
