| L(s) = 1 | − 2·5-s − 3·9-s − 2·13-s − 17-s − 4·19-s + 4·23-s − 25-s − 6·29-s − 4·31-s + 2·37-s + 6·41-s − 4·43-s + 6·45-s − 6·53-s − 12·59-s − 10·61-s + 4·65-s − 4·67-s − 4·71-s + 6·73-s + 12·79-s + 9·81-s − 4·83-s + 2·85-s − 10·89-s + 8·95-s − 2·97-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 9-s − 0.554·13-s − 0.242·17-s − 0.917·19-s + 0.834·23-s − 1/5·25-s − 1.11·29-s − 0.718·31-s + 0.328·37-s + 0.937·41-s − 0.609·43-s + 0.894·45-s − 0.824·53-s − 1.56·59-s − 1.28·61-s + 0.496·65-s − 0.488·67-s − 0.474·71-s + 0.702·73-s + 1.35·79-s + 81-s − 0.439·83-s + 0.216·85-s − 1.05·89-s + 0.820·95-s − 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 53312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.98376818011324, −14.67652030345076, −13.90025087775239, −13.50291622937111, −12.78603715547769, −12.39385392131390, −11.92046132854287, −11.25642480802247, −10.96646977280435, −10.60706999965465, −9.582780906964060, −9.264229860297618, −8.744638842722893, −8.005443599350900, −7.788655604463327, −7.126079170619506, −6.502631621726154, −5.900790565010751, −5.322374664086590, −4.657501630557918, −4.117945284481163, −3.460754549714821, −2.862762107733931, −2.212861659701392, −1.346507847854587, 0, 0,
1.346507847854587, 2.212861659701392, 2.862762107733931, 3.460754549714821, 4.117945284481163, 4.657501630557918, 5.322374664086590, 5.900790565010751, 6.502631621726154, 7.126079170619506, 7.788655604463327, 8.005443599350900, 8.744638842722893, 9.264229860297618, 9.582780906964060, 10.60706999965465, 10.96646977280435, 11.25642480802247, 11.92046132854287, 12.39385392131390, 12.78603715547769, 13.50291622937111, 13.90025087775239, 14.67652030345076, 14.98376818011324
