| L(s) = 1 | − 3-s + 5-s + 2·7-s + 9-s + 6·11-s + 13-s − 15-s − 17-s + 2·19-s − 2·21-s − 6·23-s + 25-s − 27-s + 9·29-s + 9·31-s − 6·33-s + 2·35-s + 12·37-s − 39-s + 6·41-s − 6·43-s + 45-s − 12·47-s − 3·49-s + 51-s + 3·53-s + 6·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s + 1.80·11-s + 0.277·13-s − 0.258·15-s − 0.242·17-s + 0.458·19-s − 0.436·21-s − 1.25·23-s + 1/5·25-s − 0.192·27-s + 1.67·29-s + 1.61·31-s − 1.04·33-s + 0.338·35-s + 1.97·37-s − 0.160·39-s + 0.937·41-s − 0.914·43-s + 0.149·45-s − 1.75·47-s − 3/7·49-s + 0.140·51-s + 0.412·53-s + 0.809·55-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
| \(L(1)\) |
\(\approx\) |
\(4.563106501\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.563106501\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 - 12 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.95583177693866, −12.48440493888128, −11.75100342504403, −11.70100995277018, −11.38013512307386, −10.73827090969989, −10.05256618196958, −9.781793478332343, −9.452508293745314, −8.602607988668209, −8.353538656100840, −7.879884892330949, −7.171605833976050, −6.477165161826785, −6.338540124021202, −6.001265468114295, −5.076363550583205, −4.774921039710906, −4.225069645101599, −3.781884615450882, −3.036319894312253, −2.273784212703334, −1.748849126986918, −0.9823946657030573, −0.7918428971755385,
0.7918428971755385, 0.9823946657030573, 1.748849126986918, 2.273784212703334, 3.036319894312253, 3.781884615450882, 4.225069645101599, 4.774921039710906, 5.076363550583205, 6.001265468114295, 6.338540124021202, 6.477165161826785, 7.171605833976050, 7.879884892330949, 8.353538656100840, 8.602607988668209, 9.452508293745314, 9.781793478332343, 10.05256618196958, 10.73827090969989, 11.38013512307386, 11.70100995277018, 11.75100342504403, 12.48440493888128, 12.95583177693866
