L(s) = 1 | + 1.80·2-s − 3-s + 1.24·4-s − 2.52·5-s − 1.80·6-s − 3.26·7-s − 1.36·8-s + 9-s − 4.54·10-s + 1.94·11-s − 1.24·12-s − 5.50·13-s − 5.88·14-s + 2.52·15-s − 4.94·16-s + 17-s + 1.80·18-s − 7.03·19-s − 3.13·20-s + 3.26·21-s + 3.49·22-s − 5.80·23-s + 1.36·24-s + 1.37·25-s − 9.90·26-s − 27-s − 4.06·28-s + ⋯ |
L(s) = 1 | + 1.27·2-s − 0.577·3-s + 0.621·4-s − 1.12·5-s − 0.735·6-s − 1.23·7-s − 0.481·8-s + 0.333·9-s − 1.43·10-s + 0.585·11-s − 0.359·12-s − 1.52·13-s − 1.57·14-s + 0.651·15-s − 1.23·16-s + 0.242·17-s + 0.424·18-s − 1.61·19-s − 0.702·20-s + 0.712·21-s + 0.745·22-s − 1.21·23-s + 0.277·24-s + 0.274·25-s − 1.94·26-s − 0.192·27-s − 0.767·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2546498858\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2546498858\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 157 | \( 1 + T \) |
good | 2 | \( 1 - 1.80T + 2T^{2} \) |
| 5 | \( 1 + 2.52T + 5T^{2} \) |
| 7 | \( 1 + 3.26T + 7T^{2} \) |
| 11 | \( 1 - 1.94T + 11T^{2} \) |
| 13 | \( 1 + 5.50T + 13T^{2} \) |
| 19 | \( 1 + 7.03T + 19T^{2} \) |
| 23 | \( 1 + 5.80T + 23T^{2} \) |
| 29 | \( 1 - 6.54T + 29T^{2} \) |
| 31 | \( 1 + 7.86T + 31T^{2} \) |
| 37 | \( 1 - 5.08T + 37T^{2} \) |
| 41 | \( 1 + 1.63T + 41T^{2} \) |
| 43 | \( 1 + 6.05T + 43T^{2} \) |
| 47 | \( 1 - 5.05T + 47T^{2} \) |
| 53 | \( 1 + 8.32T + 53T^{2} \) |
| 59 | \( 1 - 3.80T + 59T^{2} \) |
| 61 | \( 1 + 9.12T + 61T^{2} \) |
| 67 | \( 1 + 2.90T + 67T^{2} \) |
| 71 | \( 1 + 10.5T + 71T^{2} \) |
| 73 | \( 1 + 2.35T + 73T^{2} \) |
| 79 | \( 1 + 15.1T + 79T^{2} \) |
| 83 | \( 1 + 11.3T + 83T^{2} \) |
| 89 | \( 1 - 8.52T + 89T^{2} \) |
| 97 | \( 1 - 12.3T + 97T^{2} \) |
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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
−7.48463825433423803581427868277, −6.97088653746153396859224254096, −6.18564449099889659976443955316, −5.89669953656389480952468738475, −4.68132043577113088756103789700, −4.44221428376645378966893113585, −3.68285014441056974424325751038, −3.06322073263092604273616614007, −2.09221575786426076515276686646, −0.19525252513482006005203500032,
0.19525252513482006005203500032, 2.09221575786426076515276686646, 3.06322073263092604273616614007, 3.68285014441056974424325751038, 4.44221428376645378966893113585, 4.68132043577113088756103789700, 5.89669953656389480952468738475, 6.18564449099889659976443955316, 6.97088653746153396859224254096, 7.48463825433423803581427868277