| L(s) = 1 | − 2-s + 4-s − 5-s + 7-s − 8-s + 10-s + 13-s − 14-s + 16-s − 6·17-s − 4·19-s − 20-s + 25-s − 26-s + 28-s − 6·29-s − 4·31-s − 32-s + 6·34-s − 35-s − 10·37-s + 4·38-s + 40-s − 6·41-s + 8·43-s + 49-s − 50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.377·7-s − 0.353·8-s + 0.316·10-s + 0.277·13-s − 0.267·14-s + 1/4·16-s − 1.45·17-s − 0.917·19-s − 0.223·20-s + 1/5·25-s − 0.196·26-s + 0.188·28-s − 1.11·29-s − 0.718·31-s − 0.176·32-s + 1.02·34-s − 0.169·35-s − 1.64·37-s + 0.648·38-s + 0.158·40-s − 0.937·41-s + 1.21·43-s + 1/7·49-s − 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8709071618\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8709071618\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−7.938641635477663873446851167408, −7.03630321936057059723073246911, −6.79969598195925903738545151338, −5.79606452427756698131635363004, −5.08242666227609501064714245554, −4.12870066205105901167203744118, −3.58893656432863481313522630589, −2.35317997503702722569985565019, −1.81256610280880981690980954011, −0.49648263582545088333778996090,
0.49648263582545088333778996090, 1.81256610280880981690980954011, 2.35317997503702722569985565019, 3.58893656432863481313522630589, 4.12870066205105901167203744118, 5.08242666227609501064714245554, 5.79606452427756698131635363004, 6.79969598195925903738545151338, 7.03630321936057059723073246911, 7.938641635477663873446851167408
