L(s) = 1 | − 2·2-s − 3-s + 2·4-s − 5-s + 2·6-s − 2·7-s + 9-s + 2·10-s − 3·11-s − 2·12-s − 6·13-s + 4·14-s + 15-s − 4·16-s + 6·17-s − 2·18-s − 2·20-s + 2·21-s + 6·22-s − 8·23-s + 25-s + 12·26-s − 27-s − 4·28-s − 7·29-s − 2·30-s − 9·31-s + ⋯ |
L(s) = 1 | − 1.41·2-s − 0.577·3-s + 4-s − 0.447·5-s + 0.816·6-s − 0.755·7-s + 1/3·9-s + 0.632·10-s − 0.904·11-s − 0.577·12-s − 1.66·13-s + 1.06·14-s + 0.258·15-s − 16-s + 1.45·17-s − 0.471·18-s − 0.447·20-s + 0.436·21-s + 1.27·22-s − 1.66·23-s + 1/5·25-s + 2.35·26-s − 0.192·27-s − 0.755·28-s − 1.29·29-s − 0.365·30-s − 1.61·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5415 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5415 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−7.64509546013751274336947767160, −7.19078649832617698857199723696, −6.10972333950169677073379330452, −5.43512225817463052550494204876, −4.58929153407142908546944770619, −3.58542115361335694023519115814, −2.55489121109052871955241219583, −1.58425695658893422808083996822, 0, 0,
1.58425695658893422808083996822, 2.55489121109052871955241219583, 3.58542115361335694023519115814, 4.58929153407142908546944770619, 5.43512225817463052550494204876, 6.10972333950169677073379330452, 7.19078649832617698857199723696, 7.64509546013751274336947767160