L(s) = 1 | + 3-s + 5-s + 7-s + 9-s + 2·13-s + 15-s − 2·17-s + 4·19-s + 21-s + 25-s + 27-s + 2·29-s + 8·31-s + 35-s − 10·37-s + 2·39-s + 6·41-s − 4·43-s + 45-s + 49-s − 2·51-s − 2·53-s + 4·57-s + 12·59-s − 14·61-s + 63-s + 2·65-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s + 0.554·13-s + 0.258·15-s − 0.485·17-s + 0.917·19-s + 0.218·21-s + 1/5·25-s + 0.192·27-s + 0.371·29-s + 1.43·31-s + 0.169·35-s − 1.64·37-s + 0.320·39-s + 0.937·41-s − 0.609·43-s + 0.149·45-s + 1/7·49-s − 0.280·51-s − 0.274·53-s + 0.529·57-s + 1.56·59-s − 1.79·61-s + 0.125·63-s + 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 203280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 203280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−13.48785704371742, −12.90047496891836, −12.30807577466964, −11.96108766117937, −11.37569458413131, −10.93582063760054, −10.40773979887711, −9.901381892748222, −9.619199929733627, −8.853327822049261, −8.603422085608144, −8.231893904105859, −7.464288731864193, −7.195749475148242, −6.540508417161907, −6.061733353131487, −5.529222362034681, −4.905234364711412, −4.484799652318305, −3.882007947044729, −3.212434922724454, −2.793766059641129, −2.171553378987851, −1.438940776817857, −1.094934370218718, 0,
1.094934370218718, 1.438940776817857, 2.171553378987851, 2.793766059641129, 3.212434922724454, 3.882007947044729, 4.484799652318305, 4.905234364711412, 5.529222362034681, 6.061733353131487, 6.540508417161907, 7.195749475148242, 7.464288731864193, 8.231893904105859, 8.603422085608144, 8.853327822049261, 9.619199929733627, 9.901381892748222, 10.40773979887711, 10.93582063760054, 11.37569458413131, 11.96108766117937, 12.30807577466964, 12.90047496891836, 13.48785704371742