| L(s) = 1 | + 3.23i·3-s + 4.23i·7-s − 7.47·9-s + 11-s − 2.23i·13-s − 6.47i·17-s − 19-s − 13.7·21-s − i·23-s − 14.4i·27-s + 1.76·29-s + 0.472·31-s + 3.23i·33-s − 11.2i·37-s + 7.23·39-s + ⋯ |
| L(s) = 1 | + 1.86i·3-s + 1.60i·7-s − 2.49·9-s + 0.301·11-s − 0.620i·13-s − 1.56i·17-s − 0.229·19-s − 2.99·21-s − 0.208i·23-s − 2.78i·27-s + 0.327·29-s + 0.0847·31-s + 0.563i·33-s − 1.84i·37-s + 1.15·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5178695796\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5178695796\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 + iT \) |
| good | 3 | \( 1 - 3.23iT - 3T^{2} \) |
| 7 | \( 1 - 4.23iT - 7T^{2} \) |
| 11 | \( 1 - T + 11T^{2} \) |
| 13 | \( 1 + 2.23iT - 13T^{2} \) |
| 17 | \( 1 + 6.47iT - 17T^{2} \) |
| 19 | \( 1 + T + 19T^{2} \) |
| 29 | \( 1 - 1.76T + 29T^{2} \) |
| 31 | \( 1 - 0.472T + 31T^{2} \) |
| 37 | \( 1 + 11.2iT - 37T^{2} \) |
| 41 | \( 1 + 5.94T + 41T^{2} \) |
| 43 | \( 1 - 2.52iT - 43T^{2} \) |
| 47 | \( 1 + 11.7iT - 47T^{2} \) |
| 53 | \( 1 - 1.23iT - 53T^{2} \) |
| 59 | \( 1 + 3.23T + 59T^{2} \) |
| 61 | \( 1 + 7.23T + 61T^{2} \) |
| 67 | \( 1 + 12.9iT - 67T^{2} \) |
| 71 | \( 1 - 10T + 71T^{2} \) |
| 73 | \( 1 + 9.47iT - 73T^{2} \) |
| 79 | \( 1 + 15.1T + 79T^{2} \) |
| 83 | \( 1 - 9iT - 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 + 6.18iT - 97T^{2} \) |
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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
−8.628034656052946554945792962446, −7.81176195672770206572186525150, −6.58074491567002751767015933388, −5.71339902458204320830556154845, −5.23340430667200712920764933102, −4.69972495676999912759056613188, −3.69119547453675912203272561849, −2.95971726648781881940859922863, −2.29679689920389995367326545886, −0.14334852157426483231730937855,
1.25716067374894913814630004074, 1.53409519405692393840856577057, 2.80007157908086279738788731897, 3.76928302661789078441648431605, 4.57723030009085768470473059861, 5.80746839239138115610532103775, 6.53118611499460404411594976792, 6.84971888942753650203412790331, 7.57151874179737805414736503620, 8.189508772160219222691767674432
