| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 8-s + 9-s − 10-s + 12-s − 6·13-s + 15-s + 16-s − 2·17-s − 18-s − 4·19-s + 20-s − 8·23-s − 24-s + 25-s + 6·26-s + 27-s − 6·29-s − 30-s − 31-s − 32-s + 2·34-s + 36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.288·12-s − 1.66·13-s + 0.258·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.917·19-s + 0.223·20-s − 1.66·23-s − 0.204·24-s + 1/5·25-s + 1.17·26-s + 0.192·27-s − 1.11·29-s − 0.182·30-s − 0.179·31-s − 0.176·32-s + 0.342·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112530 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112530 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−14.11871677759376, −13.81796118892042, −12.93902299221392, −12.78051056711781, −12.16725907185537, −11.69859958412252, −11.11416103921455, −10.49221401761299, −10.11399987137003, −9.646263948443566, −9.326636762236460, −8.687042104665306, −8.284123082000356, −7.596903132350419, −7.411546077308968, −6.664827973408340, −6.225302989250949, −5.651506027385853, −4.843963085247037, −4.501122947255844, −3.678830664651711, −3.089725796060743, −2.318727369647074, −2.002518392015173, −1.482218497014808, 0, 0,
1.482218497014808, 2.002518392015173, 2.318727369647074, 3.089725796060743, 3.678830664651711, 4.501122947255844, 4.843963085247037, 5.651506027385853, 6.225302989250949, 6.664827973408340, 7.411546077308968, 7.596903132350419, 8.284123082000356, 8.687042104665306, 9.326636762236460, 9.646263948443566, 10.11399987137003, 10.49221401761299, 11.11416103921455, 11.69859958412252, 12.16725907185537, 12.78051056711781, 12.93902299221392, 13.81796118892042, 14.11871677759376
