L(s) = 1 | − 2-s + 3-s + 4-s − 0.801·5-s − 6-s + 7-s − 8-s + 9-s + 0.801·10-s + 6.04·11-s + 12-s − 14-s − 0.801·15-s + 16-s − 2.85·17-s − 18-s − 0.939·19-s − 0.801·20-s + 21-s − 6.04·22-s + 0.911·23-s − 24-s − 4.35·25-s + 27-s + 28-s − 3.35·29-s + 0.801·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.358·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s + 0.253·10-s + 1.82·11-s + 0.288·12-s − 0.267·14-s − 0.207·15-s + 0.250·16-s − 0.691·17-s − 0.235·18-s − 0.215·19-s − 0.179·20-s + 0.218·21-s − 1.28·22-s + 0.190·23-s − 0.204·24-s − 0.871·25-s + 0.192·27-s + 0.188·28-s − 0.623·29-s + 0.146·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.958457051\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.958457051\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 + 0.801T + 5T^{2} \) |
| 11 | \( 1 - 6.04T + 11T^{2} \) |
| 17 | \( 1 + 2.85T + 17T^{2} \) |
| 19 | \( 1 + 0.939T + 19T^{2} \) |
| 23 | \( 1 - 0.911T + 23T^{2} \) |
| 29 | \( 1 + 3.35T + 29T^{2} \) |
| 31 | \( 1 + 2.15T + 31T^{2} \) |
| 37 | \( 1 - 5.91T + 37T^{2} \) |
| 41 | \( 1 - 1.19T + 41T^{2} \) |
| 43 | \( 1 - 4.07T + 43T^{2} \) |
| 47 | \( 1 + 1.27T + 47T^{2} \) |
| 53 | \( 1 - 6.15T + 53T^{2} \) |
| 59 | \( 1 - 7.29T + 59T^{2} \) |
| 61 | \( 1 + T + 61T^{2} \) |
| 67 | \( 1 + 1.35T + 67T^{2} \) |
| 71 | \( 1 - 7.78T + 71T^{2} \) |
| 73 | \( 1 - 14.4T + 73T^{2} \) |
| 79 | \( 1 - 0.131T + 79T^{2} \) |
| 83 | \( 1 - 16.9T + 83T^{2} \) |
| 89 | \( 1 - 12.7T + 89T^{2} \) |
| 97 | \( 1 - 4.85T + 97T^{2} \) |
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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
−7.984230570730050319614514848027, −7.40979811126708779083118333081, −6.66359854412158551945597449368, −6.14747577721946143360123488498, −5.05608626622557382602051467361, −3.95098248192878096664171385746, −3.79684460288115990809844922633, −2.47874741625469250918563415285, −1.76209843148887548007704271285, −0.794212686988843644058002262792,
0.794212686988843644058002262792, 1.76209843148887548007704271285, 2.47874741625469250918563415285, 3.79684460288115990809844922633, 3.95098248192878096664171385746, 5.05608626622557382602051467361, 6.14747577721946143360123488498, 6.66359854412158551945597449368, 7.40979811126708779083118333081, 7.984230570730050319614514848027