| L(s) = 1 | − 5-s + 11-s − 7·19-s + 6·23-s + 25-s − 7·29-s + 31-s − 2·37-s + 9·41-s − 6·43-s − 2·47-s − 7·49-s − 55-s + 3·59-s − 10·61-s − 2·67-s + 71-s + 4·79-s − 6·83-s − 7·89-s + 7·95-s + 2·97-s − 9·101-s − 6·103-s − 2·107-s + 3·109-s − 6·115-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.301·11-s − 1.60·19-s + 1.25·23-s + 1/5·25-s − 1.29·29-s + 0.179·31-s − 0.328·37-s + 1.40·41-s − 0.914·43-s − 0.291·47-s − 49-s − 0.134·55-s + 0.390·59-s − 1.28·61-s − 0.244·67-s + 0.118·71-s + 0.450·79-s − 0.658·83-s − 0.741·89-s + 0.718·95-s + 0.203·97-s − 0.895·101-s − 0.591·103-s − 0.193·107-s + 0.287·109-s − 0.559·115-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 7 T + p T^{2} \) |
| 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
−8.317365240674473068336982570356, −7.54227626073568434476320254549, −6.79181964211011474267484400142, −6.12399482700722823913761483281, −5.13643882139048242593994299280, −4.34897160668441502760625912840, −3.59224386547074181210339138346, −2.59389192335401374823574987580, −1.46513944669369933280069977857, 0,
1.46513944669369933280069977857, 2.59389192335401374823574987580, 3.59224386547074181210339138346, 4.34897160668441502760625912840, 5.13643882139048242593994299280, 6.12399482700722823913761483281, 6.79181964211011474267484400142, 7.54227626073568434476320254549, 8.317365240674473068336982570356
