| L(s) = 1 | − 2-s + 4-s − 5-s − 8-s + 9-s + 10-s + 16-s + 2·17-s − 18-s − 20-s + 25-s − 29-s − 32-s − 2·34-s + 36-s + 2·37-s + 40-s − 45-s − 49-s − 50-s + 58-s + 64-s + 2·68-s − 72-s − 2·73-s − 2·74-s − 80-s + ⋯ |
| L(s) = 1 | − 2-s + 4-s − 5-s − 8-s + 9-s + 10-s + 16-s + 2·17-s − 18-s − 20-s + 25-s − 29-s − 32-s − 2·34-s + 36-s + 2·37-s + 40-s − 45-s − 49-s − 50-s + 58-s + 64-s + 2·68-s − 72-s − 2·73-s − 2·74-s − 80-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 580 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 580 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5680054292\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5680054292\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 3 | \( ( 1 - T )( 1 + T ) \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( ( 1 - T )^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( ( 1 - T )^{2} \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 + T )^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 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
−10.81539284750375827913230940593, −9.957890858742709535630352227905, −9.299734628084661753224857786227, −8.014701849402231805217648020722, −7.68276332672375064854797600903, −6.78791286064728920292159904732, −5.57188153400065765493474575180, −4.12322181678785183171852145682, −3.04358024235575159745522113273, −1.27890524100705199735483507800,
1.27890524100705199735483507800, 3.04358024235575159745522113273, 4.12322181678785183171852145682, 5.57188153400065765493474575180, 6.78791286064728920292159904732, 7.68276332672375064854797600903, 8.014701849402231805217648020722, 9.299734628084661753224857786227, 9.957890858742709535630352227905, 10.81539284750375827913230940593
