| L(s) = 1 | − 2-s − 0.184·3-s + 4-s − 1.70·5-s + 0.184·6-s − 8-s − 0.965·9-s + 1.70·10-s − 1.47·11-s − 0.184·12-s − 0.547·13-s + 0.313·15-s + 16-s + 1.86·17-s + 0.965·18-s − 1.70·20-s + 1.47·22-s − 1.86·23-s + 0.184·24-s + 1.89·25-s + 0.547·26-s + 0.362·27-s − 1.20·29-s − 0.313·30-s − 32-s + 0.272·33-s − 1.86·34-s + ⋯ |
| L(s) = 1 | − 2-s − 0.184·3-s + 4-s − 1.70·5-s + 0.184·6-s − 8-s − 0.965·9-s + 1.70·10-s − 1.47·11-s − 0.184·12-s − 0.547·13-s + 0.313·15-s + 16-s + 1.86·17-s + 0.965·18-s − 1.70·20-s + 1.47·22-s − 1.86·23-s + 0.184·24-s + 1.89·25-s + 0.547·26-s + 0.362·27-s − 1.20·29-s − 0.313·30-s − 32-s + 0.272·33-s − 1.86·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2900700035\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2900700035\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 701 | \( 1 - T \) |
| good | 3 | \( 1 + 0.184T + T^{2} \) |
| 5 | \( 1 + 1.70T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 + 1.47T + T^{2} \) |
| 13 | \( 1 + 0.547T + T^{2} \) |
| 17 | \( 1 - 1.86T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + 1.86T + T^{2} \) |
| 29 | \( 1 + 1.20T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - 0.891T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - 1.20T + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - 1.96T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + 1.47T + T^{2} \) |
| 71 | \( 1 - 1.70T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + 0.891T + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + 1.96T + T^{2} \) |
| 97 | \( 1 + 1.20T + T^{2} \) |
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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
−8.716578791954896094156642564558, −8.084008994411974024173462710620, −7.66750259778070443078984294160, −7.23877336028403460529080095940, −5.82052511303299238166400346500, −5.41738824610524868708950883680, −4.03332694141702430091452542671, −3.19787900908226755241798301228, −2.36670539411092366513667299140, −0.53828749899104840074917411452,
0.53828749899104840074917411452, 2.36670539411092366513667299140, 3.19787900908226755241798301228, 4.03332694141702430091452542671, 5.41738824610524868708950883680, 5.82052511303299238166400346500, 7.23877336028403460529080095940, 7.66750259778070443078984294160, 8.084008994411974024173462710620, 8.716578791954896094156642564558
