| L(s) = 1 | + 2·5-s + 6·13-s + 6·17-s + 19-s + 4·23-s − 25-s − 2·29-s − 8·31-s − 10·37-s + 2·41-s + 4·43-s + 12·47-s − 7·49-s + 6·53-s − 12·59-s − 2·61-s + 12·65-s + 4·67-s + 10·73-s + 16·83-s + 12·85-s + 2·89-s + 2·95-s + 10·97-s + 10·101-s − 8·103-s + 4·107-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 1.66·13-s + 1.45·17-s + 0.229·19-s + 0.834·23-s − 1/5·25-s − 0.371·29-s − 1.43·31-s − 1.64·37-s + 0.312·41-s + 0.609·43-s + 1.75·47-s − 49-s + 0.824·53-s − 1.56·59-s − 0.256·61-s + 1.48·65-s + 0.488·67-s + 1.17·73-s + 1.75·83-s + 1.30·85-s + 0.211·89-s + 0.205·95-s + 1.01·97-s + 0.995·101-s − 0.788·103-s + 0.386·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.506398236\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.506398236\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−9.092693734190788229037074202448, −8.062008804214737453511932083588, −7.33667434445098454643296446245, −6.39584814801769229586264060019, −5.69178884171323828176864961556, −5.22523151161492305197654101068, −3.84859941092912106278451327058, −3.25148783575260438600086430315, −1.95255670636420425042320954209, −1.07628396925620245570845051253,
1.07628396925620245570845051253, 1.95255670636420425042320954209, 3.25148783575260438600086430315, 3.84859941092912106278451327058, 5.22523151161492305197654101068, 5.69178884171323828176864961556, 6.39584814801769229586264060019, 7.33667434445098454643296446245, 8.062008804214737453511932083588, 9.092693734190788229037074202448
