| L(s) = 1 | + 2·5-s + 4·11-s − 2·13-s − 6·17-s − 8·19-s − 25-s − 6·29-s − 8·31-s − 2·37-s + 2·41-s − 4·43-s − 8·47-s − 6·53-s + 8·55-s + 6·61-s − 4·65-s − 4·67-s + 8·71-s − 10·73-s + 16·79-s + 8·83-s − 12·85-s − 6·89-s − 16·95-s + 6·97-s + 2·101-s + 16·103-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 1.20·11-s − 0.554·13-s − 1.45·17-s − 1.83·19-s − 1/5·25-s − 1.11·29-s − 1.43·31-s − 0.328·37-s + 0.312·41-s − 0.609·43-s − 1.16·47-s − 0.824·53-s + 1.07·55-s + 0.768·61-s − 0.496·65-s − 0.488·67-s + 0.949·71-s − 1.17·73-s + 1.80·79-s + 0.878·83-s − 1.30·85-s − 0.635·89-s − 1.64·95-s + 0.609·97-s + 0.199·101-s + 1.57·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−8.373309408931680629637484917349, −7.32119224555753352111842303305, −6.52917361486339326450126089747, −6.17908275353188198229198176353, −5.16177791023533130836977920691, −4.33405029272495229169198202320, −3.58097051005902557419651238987, −2.16319834122457639545883386464, −1.80766814068560455754339816128, 0,
1.80766814068560455754339816128, 2.16319834122457639545883386464, 3.58097051005902557419651238987, 4.33405029272495229169198202320, 5.16177791023533130836977920691, 6.17908275353188198229198176353, 6.52917361486339326450126089747, 7.32119224555753352111842303305, 8.373309408931680629637484917349
