L(s) = 1 | + 0.184·2-s − 0.965·4-s − 1.70·5-s − 0.362·8-s + 9-s − 0.313·10-s + 1.47·13-s + 0.898·16-s + 0.184·18-s + 1.86·19-s + 1.64·20-s + 1.89·25-s + 0.272·26-s − 1.20·29-s − 0.547·31-s + 0.528·32-s − 0.965·36-s + 0.344·38-s + 0.616·40-s − 1.20·43-s − 1.70·45-s + 49-s + 0.349·50-s − 1.42·52-s + 1.86·53-s − 0.222·58-s + 0.891·59-s + ⋯ |
L(s) = 1 | + 0.184·2-s − 0.965·4-s − 1.70·5-s − 0.362·8-s + 9-s − 0.313·10-s + 1.47·13-s + 0.898·16-s + 0.184·18-s + 1.86·19-s + 1.64·20-s + 1.89·25-s + 0.272·26-s − 1.20·29-s − 0.547·31-s + 0.528·32-s − 0.965·36-s + 0.344·38-s + 0.616·40-s − 1.20·43-s − 1.70·45-s + 49-s + 0.349·50-s − 1.42·52-s + 1.86·53-s − 0.222·58-s + 0.891·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 991 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 991 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7573360792\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7573360792\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 991 | \( 1 - T \) |
good | 2 | \( 1 - 0.184T + T^{2} \) |
| 3 | \( 1 - T^{2} \) |
| 5 | \( 1 + 1.70T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - 1.47T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - 1.86T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + 1.20T + T^{2} \) |
| 31 | \( 1 + 0.547T + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + 1.20T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - 1.86T + T^{2} \) |
| 59 | \( 1 - 0.891T + T^{2} \) |
| 61 | \( 1 + 0.547T + T^{2} \) |
| 67 | \( 1 + 1.96T + T^{2} \) |
| 71 | \( 1 - 0.891T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - 1.47T + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 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
−10.17581786195164575002445841326, −9.214492255098261221043960027269, −8.512789834374925682177088613622, −7.67900107808892814728105196014, −7.10535515212901342277796486044, −5.70289115884169765185515481562, −4.74618932389969824550288676246, −3.75497064349743364686384942346, −3.53013791555013670843085330357, −1.06466843353885040475804140865,
1.06466843353885040475804140865, 3.53013791555013670843085330357, 3.75497064349743364686384942346, 4.74618932389969824550288676246, 5.70289115884169765185515481562, 7.10535515212901342277796486044, 7.67900107808892814728105196014, 8.512789834374925682177088613622, 9.214492255098261221043960027269, 10.17581786195164575002445841326