| L(s) = 1 | − 2-s + 4-s − 5-s + 2.91·7-s − 8-s + 10-s + 1.04·11-s + 0.0128·13-s − 2.91·14-s + 16-s − 7.56·17-s + 6.06·19-s − 20-s − 1.04·22-s + 3.93·23-s + 25-s − 0.0128·26-s + 2.91·28-s + 6.61·29-s − 2.53·31-s − 32-s + 7.56·34-s − 2.91·35-s − 1.14·37-s − 6.06·38-s + 40-s − 5.10·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.447·5-s + 1.10·7-s − 0.353·8-s + 0.316·10-s + 0.313·11-s + 0.00356·13-s − 0.778·14-s + 0.250·16-s − 1.83·17-s + 1.39·19-s − 0.223·20-s − 0.221·22-s + 0.820·23-s + 0.200·25-s − 0.00251·26-s + 0.550·28-s + 1.22·29-s − 0.455·31-s − 0.176·32-s + 1.29·34-s − 0.492·35-s − 0.188·37-s − 0.983·38-s + 0.158·40-s − 0.796·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3870 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3870 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.417250041\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.417250041\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| good | 7 | \( 1 - 2.91T + 7T^{2} \) |
| 11 | \( 1 - 1.04T + 11T^{2} \) |
| 13 | \( 1 - 0.0128T + 13T^{2} \) |
| 17 | \( 1 + 7.56T + 17T^{2} \) |
| 19 | \( 1 - 6.06T + 19T^{2} \) |
| 23 | \( 1 - 3.93T + 23T^{2} \) |
| 29 | \( 1 - 6.61T + 29T^{2} \) |
| 31 | \( 1 + 2.53T + 31T^{2} \) |
| 37 | \( 1 + 1.14T + 37T^{2} \) |
| 41 | \( 1 + 5.10T + 41T^{2} \) |
| 47 | \( 1 + 2.10T + 47T^{2} \) |
| 53 | \( 1 - 8.00T + 53T^{2} \) |
| 59 | \( 1 + 4.07T + 59T^{2} \) |
| 61 | \( 1 - 11.5T + 61T^{2} \) |
| 67 | \( 1 - 9.55T + 67T^{2} \) |
| 71 | \( 1 - 11.8T + 71T^{2} \) |
| 73 | \( 1 + 15.0T + 73T^{2} \) |
| 79 | \( 1 - 10.5T + 79T^{2} \) |
| 83 | \( 1 - 10.4T + 83T^{2} \) |
| 89 | \( 1 - 0.720T + 89T^{2} \) |
| 97 | \( 1 + 1.03T + 97T^{2} \) |
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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
−8.454070220796847403238329050353, −7.929531973205337367886262588979, −7.00948414305427014851288628330, −6.65915856008575974626131711767, −5.37883545857520142018003847936, −4.77655835966300932354511541680, −3.86627967842816558903368951775, −2.78855550212003472201400598205, −1.81059597674693844159492574316, −0.78835179387016024002027523859,
0.78835179387016024002027523859, 1.81059597674693844159492574316, 2.78855550212003472201400598205, 3.86627967842816558903368951775, 4.77655835966300932354511541680, 5.37883545857520142018003847936, 6.65915856008575974626131711767, 7.00948414305427014851288628330, 7.929531973205337367886262588979, 8.454070220796847403238329050353
