| L(s) = 1 | − 3-s − 2·4-s − 5-s − 7-s − 2·9-s + 3·11-s + 2·12-s − 5·13-s + 15-s + 4·16-s − 3·17-s − 2·19-s + 2·20-s + 21-s − 6·23-s + 25-s + 5·27-s + 2·28-s − 3·29-s − 4·31-s − 3·33-s + 35-s + 4·36-s + 2·37-s + 5·39-s − 10·43-s − 6·44-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s − 0.447·5-s − 0.377·7-s − 2/3·9-s + 0.904·11-s + 0.577·12-s − 1.38·13-s + 0.258·15-s + 16-s − 0.727·17-s − 0.458·19-s + 0.447·20-s + 0.218·21-s − 1.25·23-s + 1/5·25-s + 0.962·27-s + 0.377·28-s − 0.557·29-s − 0.718·31-s − 0.522·33-s + 0.169·35-s + 2/3·36-s + 0.328·37-s + 0.800·39-s − 1.52·43-s − 0.904·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58835 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58835 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 41 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
| 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
−14.61478340392634, −14.25230170280462, −13.48455125280384, −13.07714754018952, −12.39348175761571, −12.20611487024252, −11.50177033161236, −11.23463737110482, −10.41887726582497, −9.935393917216482, −9.410274877837081, −9.044822416844772, −8.310965598426057, −8.002434224761507, −7.255192509437189, −6.549156683494704, −6.220290782772069, −5.477106695543866, −4.851488399365523, −4.571689625293214, −3.707354031760159, −3.405672214752647, −2.408496029809493, −1.682361435923316, −0.4955358127107946, 0,
0.4955358127107946, 1.682361435923316, 2.408496029809493, 3.405672214752647, 3.707354031760159, 4.571689625293214, 4.851488399365523, 5.477106695543866, 6.220290782772069, 6.549156683494704, 7.255192509437189, 8.002434224761507, 8.310965598426057, 9.044822416844772, 9.410274877837081, 9.935393917216482, 10.41887726582497, 11.23463737110482, 11.50177033161236, 12.20611487024252, 12.39348175761571, 13.07714754018952, 13.48455125280384, 14.25230170280462, 14.61478340392634
