| L(s) = 1 | + i·3-s − 9-s − 4·11-s + 2i·13-s − 2i·17-s − 4·19-s − 8i·23-s − i·27-s + 6·29-s + 8·31-s − 4i·33-s + 6i·37-s − 2·39-s − 6·41-s − 4i·43-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 0.333·9-s − 1.20·11-s + 0.554i·13-s − 0.485i·17-s − 0.917·19-s − 1.66i·23-s − 0.192i·27-s + 1.11·29-s + 1.43·31-s − 0.696i·33-s + 0.986i·37-s − 0.320·39-s − 0.937·41-s − 0.609i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.507780954\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.507780954\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + 8iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 2iT - 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - 10iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−8.482869050102263211313531019054, −7.76738980779181533128072151681, −6.69108807978462127774149363164, −6.31277083125894241879992294326, −5.12416914069215578975935401843, −4.74447841856728311489763484640, −3.93918833559908021470805200419, −2.80171671556360785553589144335, −2.29913613008558667759988059382, −0.65232441773682362223888611681,
0.66426287377050287659223418108, 1.89406437565412581366063044597, 2.72164673517412790442608165119, 3.54897778581695383335689555321, 4.62616476056665332402059815050, 5.39710295019085920571810835142, 6.04729437280323907624897614749, 6.81176471181065967300830837226, 7.63051559102022815183868598274, 8.130590170972972603076107966712
