| L(s) = 1 | − 8·2-s + 57·3-s + 64·4-s − 456·6-s − 1.17e3·7-s − 512·8-s + 1.06e3·9-s − 7.56e3·11-s + 3.64e3·12-s + 5.37e3·13-s + 9.39e3·14-s + 4.09e3·16-s + 2.40e4·17-s − 8.49e3·18-s − 5.12e4·19-s − 6.69e4·21-s + 6.05e4·22-s − 5.76e4·23-s − 2.91e4·24-s − 4.29e4·26-s − 6.41e4·27-s − 7.51e4·28-s + 4.70e4·29-s − 1.92e5·31-s − 3.27e4·32-s − 4.31e5·33-s − 1.92e5·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.21·3-s + 1/2·4-s − 0.861·6-s − 1.29·7-s − 0.353·8-s + 0.485·9-s − 1.71·11-s + 0.609·12-s + 0.678·13-s + 0.914·14-s + 1/4·16-s + 1.18·17-s − 0.343·18-s − 1.71·19-s − 1.57·21-s + 1.21·22-s − 0.987·23-s − 0.430·24-s − 0.479·26-s − 0.626·27-s − 0.646·28-s + 0.358·29-s − 1.15·31-s − 0.176·32-s − 2.08·33-s − 0.838·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{3} T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 19 p T + p^{7} T^{2} \) |
| 7 | \( 1 + 1174 T + p^{7} T^{2} \) |
| 11 | \( 1 + 7563 T + p^{7} T^{2} \) |
| 13 | \( 1 - 5372 T + p^{7} T^{2} \) |
| 17 | \( 1 - 1413 p T + p^{7} T^{2} \) |
| 19 | \( 1 + 51235 T + p^{7} T^{2} \) |
| 23 | \( 1 + 57618 T + p^{7} T^{2} \) |
| 29 | \( 1 - 47040 T + p^{7} T^{2} \) |
| 31 | \( 1 + 192358 T + p^{7} T^{2} \) |
| 37 | \( 1 - 197066 T + p^{7} T^{2} \) |
| 41 | \( 1 + 237723 T + p^{7} T^{2} \) |
| 43 | \( 1 - 653012 T + p^{7} T^{2} \) |
| 47 | \( 1 + 826884 T + p^{7} T^{2} \) |
| 53 | \( 1 - 569022 T + p^{7} T^{2} \) |
| 59 | \( 1 - 1501080 T + p^{7} T^{2} \) |
| 61 | \( 1 + 2068918 T + p^{7} T^{2} \) |
| 67 | \( 1 + 3444349 T + p^{7} T^{2} \) |
| 71 | \( 1 - 4121052 T + p^{7} T^{2} \) |
| 73 | \( 1 + 83653 T + p^{7} T^{2} \) |
| 79 | \( 1 - 1454030 T + p^{7} T^{2} \) |
| 83 | \( 1 - 1626567 T + p^{7} T^{2} \) |
| 89 | \( 1 - 6004335 T + p^{7} T^{2} \) |
| 97 | \( 1 - 3411746 T + p^{7} T^{2} \) |
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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
−13.41761568542806010633788413873, −12.60731619877358219523707050363, −10.64196432874704504058297765566, −9.740324881803944897833814100329, −8.552170341554071007597779881179, −7.67178815733294094226552792323, −6.03004549868189627898614021450, −3.46686012602083557419523813701, −2.33756145997493763629970474635, 0,
2.33756145997493763629970474635, 3.46686012602083557419523813701, 6.03004549868189627898614021450, 7.67178815733294094226552792323, 8.552170341554071007597779881179, 9.740324881803944897833814100329, 10.64196432874704504058297765566, 12.60731619877358219523707050363, 13.41761568542806010633788413873
