L(s) = 1 | + 2.13·2-s + 2.23·3-s + 2.54·4-s + 5-s + 4.75·6-s + 1.48·7-s + 1.16·8-s + 1.97·9-s + 2.13·10-s + 4.68·11-s + 5.68·12-s − 0.361·13-s + 3.16·14-s + 2.23·15-s − 2.60·16-s − 5.47·17-s + 4.21·18-s + 2.54·20-s + 3.31·21-s + 9.98·22-s − 6.16·23-s + 2.60·24-s + 25-s − 0.769·26-s − 2.28·27-s + 3.78·28-s − 0.895·29-s + ⋯ |
L(s) = 1 | + 1.50·2-s + 1.28·3-s + 1.27·4-s + 0.447·5-s + 1.94·6-s + 0.561·7-s + 0.412·8-s + 0.658·9-s + 0.674·10-s + 1.41·11-s + 1.63·12-s − 0.100·13-s + 0.846·14-s + 0.575·15-s − 0.651·16-s − 1.32·17-s + 0.992·18-s + 0.569·20-s + 0.723·21-s + 2.12·22-s − 1.28·23-s + 0.531·24-s + 0.200·25-s − 0.150·26-s − 0.440·27-s + 0.715·28-s − 0.166·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(6.685397304\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.685397304\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 19 | \( 1 \) |
good | 2 | \( 1 - 2.13T + 2T^{2} \) |
| 3 | \( 1 - 2.23T + 3T^{2} \) |
| 7 | \( 1 - 1.48T + 7T^{2} \) |
| 11 | \( 1 - 4.68T + 11T^{2} \) |
| 13 | \( 1 + 0.361T + 13T^{2} \) |
| 17 | \( 1 + 5.47T + 17T^{2} \) |
| 23 | \( 1 + 6.16T + 23T^{2} \) |
| 29 | \( 1 + 0.895T + 29T^{2} \) |
| 31 | \( 1 - 4.80T + 31T^{2} \) |
| 37 | \( 1 - 11.3T + 37T^{2} \) |
| 41 | \( 1 + 5.23T + 41T^{2} \) |
| 43 | \( 1 - 7.21T + 43T^{2} \) |
| 47 | \( 1 + 10.8T + 47T^{2} \) |
| 53 | \( 1 + 6.52T + 53T^{2} \) |
| 59 | \( 1 + 9.80T + 59T^{2} \) |
| 61 | \( 1 - 2.34T + 61T^{2} \) |
| 67 | \( 1 - 8.85T + 67T^{2} \) |
| 71 | \( 1 - 6.41T + 71T^{2} \) |
| 73 | \( 1 - 2.86T + 73T^{2} \) |
| 79 | \( 1 - 2.06T + 79T^{2} \) |
| 83 | \( 1 - 6.16T + 83T^{2} \) |
| 89 | \( 1 + 3.32T + 89T^{2} \) |
| 97 | \( 1 - 4.71T + 97T^{2} \) |
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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
−9.242065144176408099695344340708, −8.480256510898219561476116456897, −7.69599695247890450117012160116, −6.49191342875412099422808115149, −6.18335260950007917186357117091, −4.86371496922669926296134262085, −4.23373490736197371543032001220, −3.50477401142903235970599189170, −2.49660725173525638422267355022, −1.78722008939380832806444898704,
1.78722008939380832806444898704, 2.49660725173525638422267355022, 3.50477401142903235970599189170, 4.23373490736197371543032001220, 4.86371496922669926296134262085, 6.18335260950007917186357117091, 6.49191342875412099422808115149, 7.69599695247890450117012160116, 8.480256510898219561476116456897, 9.242065144176408099695344340708