| L(s) = 1 | − 2-s − 3-s + 4-s − 2·5-s + 6-s − 8-s + 9-s + 2·10-s − 4·11-s − 12-s + 2·15-s + 16-s + 4·17-s − 18-s + 4·19-s − 2·20-s + 4·22-s − 4·23-s + 24-s − 25-s − 27-s − 2·29-s − 2·30-s − 4·31-s − 32-s + 4·33-s − 4·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.632·10-s − 1.20·11-s − 0.288·12-s + 0.516·15-s + 1/4·16-s + 0.970·17-s − 0.235·18-s + 0.917·19-s − 0.447·20-s + 0.852·22-s − 0.834·23-s + 0.204·24-s − 1/5·25-s − 0.192·27-s − 0.371·29-s − 0.365·30-s − 0.718·31-s − 0.176·32-s + 0.696·33-s − 0.685·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 167 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 12 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−15.37516423171854, −14.70921596900015, −14.02440027598370, −13.43636089282763, −12.95025839124480, −12.16546743013215, −11.79596737847852, −11.70504415397719, −10.75625220717807, −10.41737460449662, −10.00025819486387, −9.461420316776768, −8.565359849664518, −8.238345238202501, −7.696719616436523, −7.168292489442455, −6.844634110650535, −5.841100213911983, −5.361242006257917, −5.047513723673154, −3.959302332771503, −3.525299902603559, −2.862266567953880, −1.909364176314135, −1.258221572813664, 0, 0,
1.258221572813664, 1.909364176314135, 2.862266567953880, 3.525299902603559, 3.959302332771503, 5.047513723673154, 5.361242006257917, 5.841100213911983, 6.844634110650535, 7.168292489442455, 7.696719616436523, 8.238345238202501, 8.565359849664518, 9.461420316776768, 10.00025819486387, 10.41737460449662, 10.75625220717807, 11.70504415397719, 11.79596737847852, 12.16546743013215, 12.95025839124480, 13.43636089282763, 14.02440027598370, 14.70921596900015, 15.37516423171854
