| L(s) = 1 | + 2-s + 1.24·3-s + 4-s − 1.80·5-s + 1.24·6-s + 8-s + 0.554·9-s − 1.80·10-s − 0.445·11-s + 1.24·12-s − 0.445·13-s − 2.24·15-s + 16-s − 1.80·17-s + 0.554·18-s − 1.80·20-s − 0.445·22-s + 1.24·23-s + 1.24·24-s + 2.24·25-s − 0.445·26-s − 0.554·27-s − 2.24·30-s + 32-s − 0.554·33-s − 1.80·34-s + 0.554·36-s + ⋯ |
| L(s) = 1 | + 2-s + 1.24·3-s + 4-s − 1.80·5-s + 1.24·6-s + 8-s + 0.554·9-s − 1.80·10-s − 0.445·11-s + 1.24·12-s − 0.445·13-s − 2.24·15-s + 16-s − 1.80·17-s + 0.554·18-s − 1.80·20-s − 0.445·22-s + 1.24·23-s + 1.24·24-s + 2.24·25-s − 0.445·26-s − 0.554·27-s − 2.24·30-s + 32-s − 0.554·33-s − 1.80·34-s + 0.554·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.696819390\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.696819390\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 67 | \( 1 - T \) |
| good | 3 | \( 1 - 1.24T + T^{2} \) |
| 5 | \( 1 + 1.80T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 + 0.445T + T^{2} \) |
| 13 | \( 1 + 0.445T + T^{2} \) |
| 17 | \( 1 + 1.80T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - 1.24T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + 1.80T + T^{2} \) |
| 47 | \( 1 + 0.445T + T^{2} \) |
| 53 | \( 1 - 1.24T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - 1.24T + T^{2} \) |
| 71 | \( 1 + 1.80T + T^{2} \) |
| 73 | \( 1 + 0.445T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - 1.24T + T^{2} \) |
| 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
−11.34330961648738925565185655446, −10.42734175119987492270858950920, −8.913052568313978534958815135326, −8.272550328650288819930158224682, −7.39871125387139687889328353702, −6.80409377554013681384434192372, −5.01867359475781226731018979655, −4.17602976255199282863717669276, −3.32976683050202977692288454573, −2.42865517235479292273872198858,
2.42865517235479292273872198858, 3.32976683050202977692288454573, 4.17602976255199282863717669276, 5.01867359475781226731018979655, 6.80409377554013681384434192372, 7.39871125387139687889328353702, 8.272550328650288819930158224682, 8.913052568313978534958815135326, 10.42734175119987492270858950920, 11.34330961648738925565185655446
