| L(s) = 1 | − 1.82·2-s + 2.33·4-s + 0.209·5-s − 2.44·8-s − 0.381·10-s + 1.95·11-s + 2.12·16-s + 17-s + 0.488·20-s − 3.57·22-s − 0.956·25-s − 1.33·29-s + 1.82·31-s − 1.44·32-s − 1.82·34-s − 0.511·40-s + 4.57·44-s + 49-s + 1.74·50-s + 0.408·55-s + 2.44·58-s − 61-s − 3.33·62-s + 0.511·64-s − 67-s + 2.33·68-s − 0.618·71-s + ⋯ |
| L(s) = 1 | − 1.82·2-s + 2.33·4-s + 0.209·5-s − 2.44·8-s − 0.381·10-s + 1.95·11-s + 2.12·16-s + 17-s + 0.488·20-s − 3.57·22-s − 0.956·25-s − 1.33·29-s + 1.82·31-s − 1.44·32-s − 1.82·34-s − 0.511·40-s + 4.57·44-s + 49-s + 1.74·50-s + 0.408·55-s + 2.44·58-s − 61-s − 3.33·62-s + 0.511·64-s − 67-s + 2.33·68-s − 0.618·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6176558049\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6176558049\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 + T \) |
| good | 2 | \( 1 + 1.82T + T^{2} \) |
| 5 | \( 1 - 0.209T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - 1.95T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + 1.33T + T^{2} \) |
| 31 | \( 1 - 1.82T + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( 1 + T + T^{2} \) |
| 71 | \( 1 + 0.618T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + 1.33T + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.218184874839161188859582030965, −8.722252403268304424275682612385, −7.82028148609667635372241623153, −7.21469475726825260660038844436, −6.36249517963208140892935974079, −5.81824088485208685177204410498, −4.23169655488536888383498136458, −3.16898213923016694376133689031, −1.90007999858281985211134067202, −1.08673799634857175466162071263,
1.08673799634857175466162071263, 1.90007999858281985211134067202, 3.16898213923016694376133689031, 4.23169655488536888383498136458, 5.81824088485208685177204410498, 6.36249517963208140892935974079, 7.21469475726825260660038844436, 7.82028148609667635372241623153, 8.722252403268304424275682612385, 9.218184874839161188859582030965
