| L(s) = 1 | − 2·5-s + 7-s − 3·9-s + 11-s + 6·13-s − 2·17-s + 8·19-s − 25-s + 29-s + 4·31-s − 2·35-s − 2·37-s − 2·41-s + 4·43-s + 6·45-s + 4·47-s + 49-s − 10·53-s − 2·55-s − 12·59-s + 2·61-s − 3·63-s − 12·65-s − 12·67-s − 8·71-s − 10·73-s + 77-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.377·7-s − 9-s + 0.301·11-s + 1.66·13-s − 0.485·17-s + 1.83·19-s − 1/5·25-s + 0.185·29-s + 0.718·31-s − 0.338·35-s − 0.328·37-s − 0.312·41-s + 0.609·43-s + 0.894·45-s + 0.583·47-s + 1/7·49-s − 1.37·53-s − 0.269·55-s − 1.56·59-s + 0.256·61-s − 0.377·63-s − 1.48·65-s − 1.46·67-s − 0.949·71-s − 1.17·73-s + 0.113·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 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 | \( 1 - 6 T + p T^{2} \) |
| 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
−15.38730096066423, −14.56760417527740, −14.13201514227066, −13.59364890728359, −13.36433739121505, −12.29020274701002, −11.97419671716612, −11.45955234471333, −11.11112054432367, −10.65320321444736, −9.816803597226805, −9.157389133723176, −8.648573697626658, −8.285943787958297, −7.598913098829343, −7.239332558302824, −6.199185767871197, −6.006724920093528, −5.225782465243876, −4.514279920914126, −3.928488684538726, −3.225765598710722, −2.874870180045167, −1.640101363621650, −1.021572557924736, 0,
1.021572557924736, 1.640101363621650, 2.874870180045167, 3.225765598710722, 3.928488684538726, 4.514279920914126, 5.225782465243876, 6.006724920093528, 6.199185767871197, 7.239332558302824, 7.598913098829343, 8.285943787958297, 8.648573697626658, 9.157389133723176, 9.816803597226805, 10.65320321444736, 11.11112054432367, 11.45955234471333, 11.97419671716612, 12.29020274701002, 13.36433739121505, 13.59364890728359, 14.13201514227066, 14.56760417527740, 15.38730096066423
