| L(s) = 1 | − 2-s + 4-s + 5-s − 8-s − 9-s − 10-s + 13-s + 16-s + 18-s + 20-s + 25-s − 26-s − 2·29-s − 32-s − 36-s − 2·37-s − 40-s − 45-s + 49-s − 50-s + 52-s + 2·58-s − 2·61-s + 64-s + 65-s + 72-s − 2·73-s + ⋯ |
| L(s) = 1 | − 2-s + 4-s + 5-s − 8-s − 9-s − 10-s + 13-s + 16-s + 18-s + 20-s + 25-s − 26-s − 2·29-s − 32-s − 36-s − 2·37-s − 40-s − 45-s + 49-s − 50-s + 52-s + 2·58-s − 2·61-s + 64-s + 65-s + 72-s − 2·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5593806808\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5593806808\) |
| \(L(1)\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + T^{2} \) |
| 7 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( 1 + T^{2} \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( ( 1 + T )^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( ( 1 + T )^{2} \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( ( 1 + T )^{2} \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( ( 1 + T )^{2} \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 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
−11.94647218702814586836801404294, −11.00483906878836862448895566271, −10.33058873652575788335465253869, −9.132478384186372660462788294960, −8.710675897752151438735129815206, −7.41880792509191377093590257542, −6.21424472545932471134617992686, −5.52349901631491846553511540766, −3.27505877239640448085289212845, −1.84288875930942106719670363354,
1.84288875930942106719670363354, 3.27505877239640448085289212845, 5.52349901631491846553511540766, 6.21424472545932471134617992686, 7.41880792509191377093590257542, 8.710675897752151438735129815206, 9.132478384186372660462788294960, 10.33058873652575788335465253869, 11.00483906878836862448895566271, 11.94647218702814586836801404294
