| L(s) = 1 | + 2·5-s − 2·7-s + 4·13-s + 2·17-s + 2·19-s − 25-s + 10·29-s − 4·35-s + 10·37-s + 6·41-s − 10·43-s − 4·47-s − 3·49-s − 2·53-s + 8·59-s − 8·61-s + 8·65-s − 4·67-s − 12·71-s + 4·73-s + 14·79-s + 12·83-s + 4·85-s − 2·89-s − 8·91-s + 4·95-s − 14·97-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.755·7-s + 1.10·13-s + 0.485·17-s + 0.458·19-s − 1/5·25-s + 1.85·29-s − 0.676·35-s + 1.64·37-s + 0.937·41-s − 1.52·43-s − 0.583·47-s − 3/7·49-s − 0.274·53-s + 1.04·59-s − 1.02·61-s + 0.992·65-s − 0.488·67-s − 1.42·71-s + 0.468·73-s + 1.57·79-s + 1.31·83-s + 0.433·85-s − 0.211·89-s − 0.838·91-s + 0.410·95-s − 1.42·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.562390121\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.562390121\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−7.899444957704501078943978226511, −6.86496647254011168098517581348, −6.27672866723048133612075007991, −5.91848300546489936729513033852, −5.07525224770060843102419515066, −4.23754731633881280955135176003, −3.31353014032490521264241817988, −2.76264700682927357741540854586, −1.68380142774966721067261344167, −0.813046206531252048775568480734,
0.813046206531252048775568480734, 1.68380142774966721067261344167, 2.76264700682927357741540854586, 3.31353014032490521264241817988, 4.23754731633881280955135176003, 5.07525224770060843102419515066, 5.91848300546489936729513033852, 6.27672866723048133612075007991, 6.86496647254011168098517581348, 7.899444957704501078943978226511
