| L(s) = 1 | + 2.11·2-s − 1.84·3-s + 2.49·4-s − 2.40·5-s − 3.90·6-s − 7-s + 1.04·8-s + 0.388·9-s − 5.10·10-s − 5.87·11-s − 4.58·12-s − 6.24·13-s − 2.11·14-s + 4.43·15-s − 2.77·16-s + 5.42·17-s + 0.823·18-s + 2.23·19-s − 6.00·20-s + 1.84·21-s − 12.4·22-s − 1.92·24-s + 0.804·25-s − 13.2·26-s + 4.80·27-s − 2.49·28-s + 0.642·29-s + ⋯ |
| L(s) = 1 | + 1.49·2-s − 1.06·3-s + 1.24·4-s − 1.07·5-s − 1.59·6-s − 0.377·7-s + 0.368·8-s + 0.129·9-s − 1.61·10-s − 1.77·11-s − 1.32·12-s − 1.73·13-s − 0.566·14-s + 1.14·15-s − 0.693·16-s + 1.31·17-s + 0.193·18-s + 0.513·19-s − 1.34·20-s + 0.401·21-s − 2.65·22-s − 0.391·24-s + 0.160·25-s − 2.59·26-s + 0.925·27-s − 0.470·28-s + 0.119·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3703 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3703 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.012014608\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.012014608\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 - 2.11T + 2T^{2} \) |
| 3 | \( 1 + 1.84T + 3T^{2} \) |
| 5 | \( 1 + 2.40T + 5T^{2} \) |
| 11 | \( 1 + 5.87T + 11T^{2} \) |
| 13 | \( 1 + 6.24T + 13T^{2} \) |
| 17 | \( 1 - 5.42T + 17T^{2} \) |
| 19 | \( 1 - 2.23T + 19T^{2} \) |
| 29 | \( 1 - 0.642T + 29T^{2} \) |
| 31 | \( 1 - 7.84T + 31T^{2} \) |
| 37 | \( 1 + 0.557T + 37T^{2} \) |
| 41 | \( 1 - 2.56T + 41T^{2} \) |
| 43 | \( 1 - 8.81T + 43T^{2} \) |
| 47 | \( 1 - 4.26T + 47T^{2} \) |
| 53 | \( 1 + 3.01T + 53T^{2} \) |
| 59 | \( 1 - 4.17T + 59T^{2} \) |
| 61 | \( 1 - 0.148T + 61T^{2} \) |
| 67 | \( 1 + 13.3T + 67T^{2} \) |
| 71 | \( 1 + 7.93T + 71T^{2} \) |
| 73 | \( 1 - 4.28T + 73T^{2} \) |
| 79 | \( 1 - 0.861T + 79T^{2} \) |
| 83 | \( 1 - 4.81T + 83T^{2} \) |
| 89 | \( 1 + 6.32T + 89T^{2} \) |
| 97 | \( 1 + 10.4T + 97T^{2} \) |
| 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
−8.078419420625043061056629504803, −7.56979782953648572371073827433, −6.91823441971361680770236101520, −5.86053440646181145362010397187, −5.42393286727756747101160253939, −4.83200494445930651942375990170, −4.18615125900385369287888021913, −2.98817161039328174894878166289, −2.68011728261128694180907430635, −0.45787216697059175790896577780,
0.45787216697059175790896577780, 2.68011728261128694180907430635, 2.98817161039328174894878166289, 4.18615125900385369287888021913, 4.83200494445930651942375990170, 5.42393286727756747101160253939, 5.86053440646181145362010397187, 6.91823441971361680770236101520, 7.56979782953648572371073827433, 8.078419420625043061056629504803
