| L(s) = 1 | + 2-s − 3-s + 4-s + 2·5-s − 6-s − 7-s + 8-s + 9-s + 2·10-s − 12-s − 2·13-s − 14-s − 2·15-s + 16-s + 2·17-s + 18-s + 4·19-s + 2·20-s + 21-s − 4·23-s − 24-s − 25-s − 2·26-s − 27-s − 28-s + 6·29-s − 2·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.632·10-s − 0.288·12-s − 0.554·13-s − 0.267·14-s − 0.516·15-s + 1/4·16-s + 0.485·17-s + 0.235·18-s + 0.917·19-s + 0.447·20-s + 0.218·21-s − 0.834·23-s − 0.204·24-s − 1/5·25-s − 0.392·26-s − 0.192·27-s − 0.188·28-s + 1.11·29-s − 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77658 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77658 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 43 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 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 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−14.08148879299621, −13.75207783805403, −13.48393575722653, −12.60768782242722, −12.36690425968793, −11.93822497645779, −11.44717710477199, −10.75305654483034, −10.24360201906109, −9.820078464603151, −9.563317037851471, −8.735921741743867, −8.008839621749998, −7.550196304262066, −6.864454082840380, −6.345051288777824, −6.050914363559470, −5.331903731644334, −5.016170759574876, −4.433034834980292, −3.591816734930581, −3.117468304105803, −2.389534659969034, −1.737589460077815, −1.054286365717681, 0,
1.054286365717681, 1.737589460077815, 2.389534659969034, 3.117468304105803, 3.591816734930581, 4.433034834980292, 5.016170759574876, 5.331903731644334, 6.050914363559470, 6.345051288777824, 6.864454082840380, 7.550196304262066, 8.008839621749998, 8.735921741743867, 9.563317037851471, 9.820078464603151, 10.24360201906109, 10.75305654483034, 11.44717710477199, 11.93822497645779, 12.36690425968793, 12.60768782242722, 13.48393575722653, 13.75207783805403, 14.08148879299621
