L(s) = 1 | − 2.39·2-s + 3-s + 3.72·4-s − 2.39·6-s + 4.15·7-s − 4.13·8-s + 9-s − 5.71·11-s + 3.72·12-s + 3.79·13-s − 9.93·14-s + 2.44·16-s − 2.66·17-s − 2.39·18-s + 19-s + 4.15·21-s + 13.6·22-s + 8.13·23-s − 4.13·24-s − 9.09·26-s + 27-s + 15.4·28-s + 4.73·29-s − 2.31·31-s + 2.42·32-s − 5.71·33-s + 6.38·34-s + ⋯ |
L(s) = 1 | − 1.69·2-s + 0.577·3-s + 1.86·4-s − 0.977·6-s + 1.56·7-s − 1.46·8-s + 0.333·9-s − 1.72·11-s + 1.07·12-s + 1.05·13-s − 2.65·14-s + 0.610·16-s − 0.646·17-s − 0.564·18-s + 0.229·19-s + 0.906·21-s + 2.91·22-s + 1.69·23-s − 0.844·24-s − 1.78·26-s + 0.192·27-s + 2.92·28-s + 0.878·29-s − 0.415·31-s + 0.428·32-s − 0.995·33-s + 1.09·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.090397187\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.090397187\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 2 | \( 1 + 2.39T + 2T^{2} \) |
| 7 | \( 1 - 4.15T + 7T^{2} \) |
| 11 | \( 1 + 5.71T + 11T^{2} \) |
| 13 | \( 1 - 3.79T + 13T^{2} \) |
| 17 | \( 1 + 2.66T + 17T^{2} \) |
| 23 | \( 1 - 8.13T + 23T^{2} \) |
| 29 | \( 1 - 4.73T + 29T^{2} \) |
| 31 | \( 1 + 2.31T + 31T^{2} \) |
| 37 | \( 1 + 4.68T + 37T^{2} \) |
| 41 | \( 1 - 10.1T + 41T^{2} \) |
| 43 | \( 1 + 1.76T + 43T^{2} \) |
| 47 | \( 1 - 7.14T + 47T^{2} \) |
| 53 | \( 1 + 3.04T + 53T^{2} \) |
| 59 | \( 1 + 0.582T + 59T^{2} \) |
| 61 | \( 1 + 9.54T + 61T^{2} \) |
| 67 | \( 1 + 1.20T + 67T^{2} \) |
| 71 | \( 1 - 1.20T + 71T^{2} \) |
| 73 | \( 1 - 11.5T + 73T^{2} \) |
| 79 | \( 1 - 5.06T + 79T^{2} \) |
| 83 | \( 1 - 1.83T + 83T^{2} \) |
| 89 | \( 1 - 3.36T + 89T^{2} \) |
| 97 | \( 1 - 0.313T + 97T^{2} \) |
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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
−9.212255645485877232424656358630, −8.735670669198710472445147998114, −8.012315527350309643498535492697, −7.66134322626100280387941106439, −6.76104683755449074158998367121, −5.42809281217360993473538606480, −4.53614947214681502907588263069, −2.92190288206840948444097653772, −2.03233052475643602027243928755, −0.987322599412682405163559365636,
0.987322599412682405163559365636, 2.03233052475643602027243928755, 2.92190288206840948444097653772, 4.53614947214681502907588263069, 5.42809281217360993473538606480, 6.76104683755449074158998367121, 7.66134322626100280387941106439, 8.012315527350309643498535492697, 8.735670669198710472445147998114, 9.212255645485877232424656358630