| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s + 2·7-s − 8-s + 9-s + 11-s − 12-s − 4·13-s − 2·14-s + 16-s + 2·17-s − 18-s − 2·21-s − 22-s + 6·23-s + 24-s + 4·26-s − 27-s + 2·28-s + 10·29-s − 8·31-s − 32-s − 33-s − 2·34-s + 36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.301·11-s − 0.288·12-s − 1.10·13-s − 0.534·14-s + 1/4·16-s + 0.485·17-s − 0.235·18-s − 0.436·21-s − 0.213·22-s + 1.25·23-s + 0.204·24-s + 0.784·26-s − 0.192·27-s + 0.377·28-s + 1.85·29-s − 1.43·31-s − 0.176·32-s − 0.174·33-s − 0.342·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.065812692\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.065812692\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−9.439951884838861027916558503561, −8.598963630436687415232742321826, −7.74784194457137525832622675815, −7.10802519408124484946137932528, −6.29830546140412783624908940380, −5.21047015697783548224554894519, −4.64555811808454330278117280236, −3.23515428831679737913841272058, −2.01647539997832019004528400005, −0.840925759728957288768944649377,
0.840925759728957288768944649377, 2.01647539997832019004528400005, 3.23515428831679737913841272058, 4.64555811808454330278117280236, 5.21047015697783548224554894519, 6.29830546140412783624908940380, 7.10802519408124484946137932528, 7.74784194457137525832622675815, 8.598963630436687415232742321826, 9.439951884838861027916558503561
