| L(s) = 1 | − 3-s − 5-s + 2·7-s + 9-s + 4·11-s − 13-s + 15-s + 17-s + 6·19-s − 2·21-s − 8·23-s + 25-s − 27-s − 6·29-s + 8·31-s − 4·33-s − 2·35-s + 2·37-s + 39-s − 2·41-s − 10·43-s − 45-s + 4·47-s − 3·49-s − 51-s + 2·53-s − 4·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s + 1.20·11-s − 0.277·13-s + 0.258·15-s + 0.242·17-s + 1.37·19-s − 0.436·21-s − 1.66·23-s + 1/5·25-s − 0.192·27-s − 1.11·29-s + 1.43·31-s − 0.696·33-s − 0.338·35-s + 0.328·37-s + 0.160·39-s − 0.312·41-s − 1.52·43-s − 0.149·45-s + 0.583·47-s − 3/7·49-s − 0.140·51-s + 0.274·53-s − 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−13.25430349800526, −12.51960418564446, −12.21946397629838, −11.71593263778448, −11.42915119052395, −11.25248402989861, −10.35881627698114, −9.972794222665075, −9.578741845388198, −9.098086264006987, −8.285811345727558, −8.074289434906987, −7.584543608452325, −6.948049041557823, −6.594647242829052, −5.986826582139088, −5.442773317163023, −5.023961167281849, −4.399733557755539, −3.956332183159878, −3.495102093505389, −2.769024375783612, −1.922918603079481, −1.438879063116220, −0.8468244130150362, 0,
0.8468244130150362, 1.438879063116220, 1.922918603079481, 2.769024375783612, 3.495102093505389, 3.956332183159878, 4.399733557755539, 5.023961167281849, 5.442773317163023, 5.986826582139088, 6.594647242829052, 6.948049041557823, 7.584543608452325, 8.074289434906987, 8.285811345727558, 9.098086264006987, 9.578741845388198, 9.972794222665075, 10.35881627698114, 11.25248402989861, 11.42915119052395, 11.71593263778448, 12.21946397629838, 12.51960418564446, 13.25430349800526
