L(s) = 1 | + 5-s + 4·11-s − 2·13-s + 2·17-s + 19-s − 8·23-s + 25-s − 6·29-s − 8·31-s − 10·37-s + 2·41-s − 8·43-s + 8·47-s − 7·49-s + 6·53-s + 4·55-s + 8·59-s − 10·61-s − 2·65-s + 4·67-s − 16·71-s − 6·73-s − 16·79-s − 4·83-s + 2·85-s − 6·89-s + 95-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 1.20·11-s − 0.554·13-s + 0.485·17-s + 0.229·19-s − 1.66·23-s + 1/5·25-s − 1.11·29-s − 1.43·31-s − 1.64·37-s + 0.312·41-s − 1.21·43-s + 1.16·47-s − 49-s + 0.824·53-s + 0.539·55-s + 1.04·59-s − 1.28·61-s − 0.248·65-s + 0.488·67-s − 1.89·71-s − 0.702·73-s − 1.80·79-s − 0.439·83-s + 0.216·85-s − 0.635·89-s + 0.102·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 19 | \( 1 - T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 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
−7.30960781775528372538223841393, −7.17042769370238463569332406995, −6.00098346347354001975619293790, −5.71990223663399066758929349280, −4.75613279233959624961003844520, −3.89431269377804145473253356551, −3.30868983127856611761853430868, −2.06871305245572464607004717841, −1.50188955546054581417174306023, 0,
1.50188955546054581417174306023, 2.06871305245572464607004717841, 3.30868983127856611761853430868, 3.89431269377804145473253356551, 4.75613279233959624961003844520, 5.71990223663399066758929349280, 6.00098346347354001975619293790, 7.17042769370238463569332406995, 7.30960781775528372538223841393