| L(s) = 1 | + i·3-s + (0.311 − 2.21i)5-s + 4.90i·7-s − 9-s + 11-s + 4.14i·13-s + (2.21 + 0.311i)15-s + 5.33i·17-s − 5.18·19-s − 4.90·21-s − 4i·23-s + (−4.80 − 1.37i)25-s − i·27-s − 1.80·29-s − 2.62·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + (0.139 − 0.990i)5-s + 1.85i·7-s − 0.333·9-s + 0.301·11-s + 1.15i·13-s + (0.571 + 0.0803i)15-s + 1.29i·17-s − 1.18·19-s − 1.06·21-s − 0.834i·23-s + (−0.961 − 0.275i)25-s − 0.192i·27-s − 0.335·29-s − 0.470·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.139i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 - 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8813955548\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8813955548\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (-0.311 + 2.21i)T \) |
| 11 | \( 1 - T \) |
| good | 7 | \( 1 - 4.90iT - 7T^{2} \) |
| 13 | \( 1 - 4.14iT - 13T^{2} \) |
| 17 | \( 1 - 5.33iT - 17T^{2} \) |
| 19 | \( 1 + 5.18T + 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 + 1.80T + 29T^{2} \) |
| 31 | \( 1 + 2.62T + 31T^{2} \) |
| 37 | \( 1 + 5.80iT - 37T^{2} \) |
| 41 | \( 1 - 1.80T + 41T^{2} \) |
| 43 | \( 1 + 4.90iT - 43T^{2} \) |
| 47 | \( 1 - 7.05iT - 47T^{2} \) |
| 53 | \( 1 + 7.18iT - 53T^{2} \) |
| 59 | \( 1 - 1.67T + 59T^{2} \) |
| 61 | \( 1 - 0.755T + 61T^{2} \) |
| 67 | \( 1 + 4.85iT - 67T^{2} \) |
| 71 | \( 1 + 0.428T + 71T^{2} \) |
| 73 | \( 1 - 12.7iT - 73T^{2} \) |
| 79 | \( 1 + 6.42T + 79T^{2} \) |
| 83 | \( 1 - 2.90iT - 83T^{2} \) |
| 89 | \( 1 + 0.622T + 89T^{2} \) |
| 97 | \( 1 - 2.75iT - 97T^{2} \) |
| 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
−9.139802904578349493196110997163, −8.600327972004237773635333529809, −8.186505105761158373685278597371, −6.67463345187058765259299951920, −5.96322073996326127071579241637, −5.40661109872207471851989109025, −4.46594944128980453040777266346, −3.86091782906617814168235339385, −2.39371914054203298518144493601, −1.77532473541198652605608151932,
0.27579647668228536578355633447, 1.46287185118712345549711804443, 2.77514270505209182235749349670, 3.52303173437102395124376152919, 4.39892290537778238486795716724, 5.51618257470267948394476853436, 6.45878661838847249585147021706, 7.07104601831783766970802977765, 7.53628579320893297039126451593, 8.190603013709827337523653058503
