| L(s) = 1 | + 0.185·3-s + 4.45·7-s − 2.96·9-s + 2.64·11-s − 1.30·13-s + 3.51·17-s − 19-s + 0.826·21-s + 6.52·23-s − 1.10·27-s − 5.20·29-s + 10.8·31-s + 0.490·33-s − 2.04·37-s − 0.241·39-s − 3.80·41-s − 4.77·43-s − 1.49·47-s + 12.8·49-s + 0.652·51-s − 0.225·53-s − 0.185·57-s − 2.86·59-s − 6.31·61-s − 13.2·63-s + 13.1·67-s + 1.21·69-s + ⋯ |
| L(s) = 1 | + 0.107·3-s + 1.68·7-s − 0.988·9-s + 0.797·11-s − 0.360·13-s + 0.853·17-s − 0.229·19-s + 0.180·21-s + 1.36·23-s − 0.212·27-s − 0.967·29-s + 1.94·31-s + 0.0854·33-s − 0.336·37-s − 0.0386·39-s − 0.593·41-s − 0.727·43-s − 0.217·47-s + 1.83·49-s + 0.0913·51-s − 0.0309·53-s − 0.0245·57-s − 0.372·59-s − 0.808·61-s − 1.66·63-s + 1.61·67-s + 0.145·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.484218993\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.484218993\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 0.185T + 3T^{2} \) |
| 7 | \( 1 - 4.45T + 7T^{2} \) |
| 11 | \( 1 - 2.64T + 11T^{2} \) |
| 13 | \( 1 + 1.30T + 13T^{2} \) |
| 17 | \( 1 - 3.51T + 17T^{2} \) |
| 23 | \( 1 - 6.52T + 23T^{2} \) |
| 29 | \( 1 + 5.20T + 29T^{2} \) |
| 31 | \( 1 - 10.8T + 31T^{2} \) |
| 37 | \( 1 + 2.04T + 37T^{2} \) |
| 41 | \( 1 + 3.80T + 41T^{2} \) |
| 43 | \( 1 + 4.77T + 43T^{2} \) |
| 47 | \( 1 + 1.49T + 47T^{2} \) |
| 53 | \( 1 + 0.225T + 53T^{2} \) |
| 59 | \( 1 + 2.86T + 59T^{2} \) |
| 61 | \( 1 + 6.31T + 61T^{2} \) |
| 67 | \( 1 - 13.1T + 67T^{2} \) |
| 71 | \( 1 - 12.3T + 71T^{2} \) |
| 73 | \( 1 - 5.42T + 73T^{2} \) |
| 79 | \( 1 - 14.9T + 79T^{2} \) |
| 83 | \( 1 + 3.84T + 83T^{2} \) |
| 89 | \( 1 - 1.67T + 89T^{2} \) |
| 97 | \( 1 + 9.48T + 97T^{2} \) |
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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
−8.272624986136945238379650887192, −8.063842296408725867187989584992, −7.08185461298803165917526987803, −6.29206888048647638496356029544, −5.23525721337534450687527097188, −4.95257414709823673884497690017, −3.88847013151906623993700021614, −2.94307679812171951738207131986, −1.92945799514007907577466530205, −0.965009873455249558456152894215,
0.965009873455249558456152894215, 1.92945799514007907577466530205, 2.94307679812171951738207131986, 3.88847013151906623993700021614, 4.95257414709823673884497690017, 5.23525721337534450687527097188, 6.29206888048647638496356029544, 7.08185461298803165917526987803, 8.063842296408725867187989584992, 8.272624986136945238379650887192
