| L(s) = 1 | + 1.65·2-s − 1.97·3-s + 0.737·4-s − 3.26·6-s − 2.24·7-s − 2.08·8-s + 0.899·9-s − 1.45·12-s − 3.69·13-s − 3.71·14-s − 4.93·16-s − 2.22·17-s + 1.48·18-s + 5.28·19-s + 4.42·21-s − 3.85·23-s + 4.12·24-s − 6.12·26-s + 4.14·27-s − 1.65·28-s + 0.188·29-s + 0.686·31-s − 3.98·32-s − 3.68·34-s + 0.663·36-s + 2.59·37-s + 8.74·38-s + ⋯ |
| L(s) = 1 | + 1.16·2-s − 1.14·3-s + 0.368·4-s − 1.33·6-s − 0.847·7-s − 0.738·8-s + 0.299·9-s − 0.420·12-s − 1.02·13-s − 0.991·14-s − 1.23·16-s − 0.539·17-s + 0.350·18-s + 1.21·19-s + 0.966·21-s − 0.803·23-s + 0.841·24-s − 1.20·26-s + 0.798·27-s − 0.312·28-s + 0.0349·29-s + 0.123·31-s − 0.703·32-s − 0.631·34-s + 0.110·36-s + 0.426·37-s + 1.41·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.109368232\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.109368232\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 1.65T + 2T^{2} \) |
| 3 | \( 1 + 1.97T + 3T^{2} \) |
| 7 | \( 1 + 2.24T + 7T^{2} \) |
| 13 | \( 1 + 3.69T + 13T^{2} \) |
| 17 | \( 1 + 2.22T + 17T^{2} \) |
| 19 | \( 1 - 5.28T + 19T^{2} \) |
| 23 | \( 1 + 3.85T + 23T^{2} \) |
| 29 | \( 1 - 0.188T + 29T^{2} \) |
| 31 | \( 1 - 0.686T + 31T^{2} \) |
| 37 | \( 1 - 2.59T + 37T^{2} \) |
| 41 | \( 1 - 7.91T + 41T^{2} \) |
| 43 | \( 1 - 8.41T + 43T^{2} \) |
| 47 | \( 1 + 12.0T + 47T^{2} \) |
| 53 | \( 1 - 12.6T + 53T^{2} \) |
| 59 | \( 1 + 0.343T + 59T^{2} \) |
| 61 | \( 1 - 1.73T + 61T^{2} \) |
| 67 | \( 1 - 0.650T + 67T^{2} \) |
| 71 | \( 1 - 4.64T + 71T^{2} \) |
| 73 | \( 1 + 8.85T + 73T^{2} \) |
| 79 | \( 1 + 7.23T + 79T^{2} \) |
| 83 | \( 1 - 3.18T + 83T^{2} \) |
| 89 | \( 1 - 9.92T + 89T^{2} \) |
| 97 | \( 1 - 2.26T + 97T^{2} \) |
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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
−8.870146286628711338609912110373, −7.66811660495990753831464835826, −6.84360758364492796409815979189, −6.15397960854221345059547004373, −5.63821408717182862189975139252, −4.92227875479514850636791271142, −4.24511460275724949214529443714, −3.25415728288556989093670153317, −2.43566478594833706154381027098, −0.53999591252574595235015351304,
0.53999591252574595235015351304, 2.43566478594833706154381027098, 3.25415728288556989093670153317, 4.24511460275724949214529443714, 4.92227875479514850636791271142, 5.63821408717182862189975139252, 6.15397960854221345059547004373, 6.84360758364492796409815979189, 7.66811660495990753831464835826, 8.870146286628711338609912110373
