L(s) = 1 | + 2.09·2-s + 3-s + 2.39·4-s + 5-s + 2.09·6-s − 1.39·7-s + 0.834·8-s + 9-s + 2.09·10-s + 3.33·11-s + 2.39·12-s + 5.70·13-s − 2.92·14-s + 15-s − 3.04·16-s − 2.75·17-s + 2.09·18-s − 3.13·19-s + 2.39·20-s − 1.39·21-s + 6.99·22-s + 6.71·23-s + 0.834·24-s + 25-s + 11.9·26-s + 27-s − 3.34·28-s + ⋯ |
L(s) = 1 | + 1.48·2-s + 0.577·3-s + 1.19·4-s + 0.447·5-s + 0.856·6-s − 0.527·7-s + 0.294·8-s + 0.333·9-s + 0.663·10-s + 1.00·11-s + 0.692·12-s + 1.58·13-s − 0.782·14-s + 0.258·15-s − 0.761·16-s − 0.667·17-s + 0.494·18-s − 0.719·19-s + 0.536·20-s − 0.304·21-s + 1.49·22-s + 1.40·23-s + 0.170·24-s + 0.200·25-s + 2.34·26-s + 0.192·27-s − 0.632·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(6.693343136\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.693343136\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 401 | \( 1 - T \) |
good | 2 | \( 1 - 2.09T + 2T^{2} \) |
| 7 | \( 1 + 1.39T + 7T^{2} \) |
| 11 | \( 1 - 3.33T + 11T^{2} \) |
| 13 | \( 1 - 5.70T + 13T^{2} \) |
| 17 | \( 1 + 2.75T + 17T^{2} \) |
| 19 | \( 1 + 3.13T + 19T^{2} \) |
| 23 | \( 1 - 6.71T + 23T^{2} \) |
| 29 | \( 1 - 3.04T + 29T^{2} \) |
| 31 | \( 1 - 9.55T + 31T^{2} \) |
| 37 | \( 1 + 3.65T + 37T^{2} \) |
| 41 | \( 1 - 3.99T + 41T^{2} \) |
| 43 | \( 1 - 1.58T + 43T^{2} \) |
| 47 | \( 1 + 10.8T + 47T^{2} \) |
| 53 | \( 1 + 5.35T + 53T^{2} \) |
| 59 | \( 1 + 3.37T + 59T^{2} \) |
| 61 | \( 1 - 12.1T + 61T^{2} \) |
| 67 | \( 1 - 3.88T + 67T^{2} \) |
| 71 | \( 1 - 11.4T + 71T^{2} \) |
| 73 | \( 1 + 10.1T + 73T^{2} \) |
| 79 | \( 1 - 2.46T + 79T^{2} \) |
| 83 | \( 1 - 5.54T + 83T^{2} \) |
| 89 | \( 1 - 14.6T + 89T^{2} \) |
| 97 | \( 1 + 13.5T + 97T^{2} \) |
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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
−8.173132123483470805815232739461, −6.80291988819102305121346631280, −6.52087383313922542937764570538, −6.09152736019509918712429837827, −4.99862748953386544207075419951, −4.39894391018011587055719721091, −3.61243446379562210891931308743, −3.11867245233017301530324432104, −2.21603337691121290145273288902, −1.13995377133954938655384621026,
1.13995377133954938655384621026, 2.21603337691121290145273288902, 3.11867245233017301530324432104, 3.61243446379562210891931308743, 4.39894391018011587055719721091, 4.99862748953386544207075419951, 6.09152736019509918712429837827, 6.52087383313922542937764570538, 6.80291988819102305121346631280, 8.173132123483470805815232739461