| L(s) = 1 | − 2-s + 1.96·3-s + 4-s + 0.342·5-s − 1.96·6-s + 2.97·7-s − 8-s + 0.842·9-s − 0.342·10-s − 11-s + 1.96·12-s + 5.18·13-s − 2.97·14-s + 0.670·15-s + 16-s − 1.22·17-s − 0.842·18-s + 0.342·20-s + 5.83·21-s + 22-s + 9.22·23-s − 1.96·24-s − 4.88·25-s − 5.18·26-s − 4.22·27-s + 2.97·28-s + 2.96·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.13·3-s + 0.5·4-s + 0.152·5-s − 0.800·6-s + 1.12·7-s − 0.353·8-s + 0.280·9-s − 0.108·10-s − 0.301·11-s + 0.565·12-s + 1.43·13-s − 0.796·14-s + 0.173·15-s + 0.250·16-s − 0.296·17-s − 0.198·18-s + 0.0764·20-s + 1.27·21-s + 0.213·22-s + 1.92·23-s − 0.400·24-s − 0.976·25-s − 1.01·26-s − 0.814·27-s + 0.563·28-s + 0.550·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.971021392\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.971021392\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 1.96T + 3T^{2} \) |
| 5 | \( 1 - 0.342T + 5T^{2} \) |
| 7 | \( 1 - 2.97T + 7T^{2} \) |
| 13 | \( 1 - 5.18T + 13T^{2} \) |
| 17 | \( 1 + 1.22T + 17T^{2} \) |
| 23 | \( 1 - 9.22T + 23T^{2} \) |
| 29 | \( 1 - 2.96T + 29T^{2} \) |
| 31 | \( 1 - 2.03T + 31T^{2} \) |
| 37 | \( 1 + 0.918T + 37T^{2} \) |
| 41 | \( 1 - 8.14T + 41T^{2} \) |
| 43 | \( 1 - 8.17T + 43T^{2} \) |
| 47 | \( 1 - 0.0493T + 47T^{2} \) |
| 53 | \( 1 + 4.78T + 53T^{2} \) |
| 59 | \( 1 + 3.40T + 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 + 3.70T + 67T^{2} \) |
| 71 | \( 1 - 16.0T + 71T^{2} \) |
| 73 | \( 1 + 6.87T + 73T^{2} \) |
| 79 | \( 1 - 3.35T + 79T^{2} \) |
| 83 | \( 1 - 15.3T + 83T^{2} \) |
| 89 | \( 1 - 6.78T + 89T^{2} \) |
| 97 | \( 1 - 11.6T + 97T^{2} \) |
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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
−7.84407808281384022498185705952, −7.65492336762539175135833764318, −6.57724954555007665820921865022, −5.89401795464774045797146547272, −5.01443822944401607825446742194, −4.15667441290628062079474722050, −3.27800706161787006971311466276, −2.57709177083257225066204230589, −1.75460470617714496374176581214, −0.957272294718045335459950003103,
0.957272294718045335459950003103, 1.75460470617714496374176581214, 2.57709177083257225066204230589, 3.27800706161787006971311466276, 4.15667441290628062079474722050, 5.01443822944401607825446742194, 5.89401795464774045797146547272, 6.57724954555007665820921865022, 7.65492336762539175135833764318, 7.84407808281384022498185705952
