| L(s) = 1 | + 2-s + 1.24·3-s + 4-s + 1.24·6-s + 8-s + 0.554·9-s − 1.80·11-s + 1.24·12-s + 16-s − 1.80·17-s + 0.554·18-s − 0.445·19-s − 1.80·22-s + 1.24·24-s + 25-s − 0.554·27-s + 32-s − 2.24·33-s − 1.80·34-s + 0.554·36-s − 0.445·38-s − 0.445·41-s − 0.445·43-s − 1.80·44-s + 1.24·48-s + 49-s + 50-s + ⋯ |
| L(s) = 1 | + 2-s + 1.24·3-s + 4-s + 1.24·6-s + 8-s + 0.554·9-s − 1.80·11-s + 1.24·12-s + 16-s − 1.80·17-s + 0.554·18-s − 0.445·19-s − 1.80·22-s + 1.24·24-s + 25-s − 0.554·27-s + 32-s − 2.24·33-s − 1.80·34-s + 0.554·36-s − 0.445·38-s − 0.445·41-s − 0.445·43-s − 1.80·44-s + 1.24·48-s + 49-s + 50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.607806684\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.607806684\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 1.24T + T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 + 1.80T + T^{2} \) |
| 17 | \( 1 + 1.80T + T^{2} \) |
| 19 | \( 1 + 0.445T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + 0.445T + T^{2} \) |
| 43 | \( 1 + 0.445T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - 1.24T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - 1.24T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - 1.24T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + 0.445T + T^{2} \) |
| 89 | \( 1 - 1.24T + T^{2} \) |
| 97 | \( 1 + 1.80T + T^{2} \) |
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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
−9.875654580850310203468841579384, −8.714175030032420517148873808864, −8.221080272590182760980213668920, −7.33649275298196193986702981684, −6.58371062562942347637027012561, −5.41492870872084987694972350263, −4.64793598838847666122462712572, −3.65656832440834897915504214923, −2.62405835445426523864152193679, −2.21165880824371858420203976660,
2.21165880824371858420203976660, 2.62405835445426523864152193679, 3.65656832440834897915504214923, 4.64793598838847666122462712572, 5.41492870872084987694972350263, 6.58371062562942347637027012561, 7.33649275298196193986702981684, 8.221080272590182760980213668920, 8.714175030032420517148873808864, 9.875654580850310203468841579384
