| L(s) = 1 | + 2·5-s + 6·11-s + 6·13-s − 2·17-s + 4·19-s + 2·23-s − 25-s − 8·29-s + 4·31-s − 6·37-s + 10·41-s + 4·43-s + 4·47-s + 4·53-s + 12·55-s + 12·59-s + 2·61-s + 12·65-s − 12·67-s + 6·71-s + 2·73-s + 8·79-s − 4·85-s − 14·89-s + 8·95-s + 2·97-s − 18·101-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 1.80·11-s + 1.66·13-s − 0.485·17-s + 0.917·19-s + 0.417·23-s − 1/5·25-s − 1.48·29-s + 0.718·31-s − 0.986·37-s + 1.56·41-s + 0.609·43-s + 0.583·47-s + 0.549·53-s + 1.61·55-s + 1.56·59-s + 0.256·61-s + 1.48·65-s − 1.46·67-s + 0.712·71-s + 0.234·73-s + 0.900·79-s − 0.433·85-s − 1.48·89-s + 0.820·95-s + 0.203·97-s − 1.79·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.282992578\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.282992578\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−7.998025692462453871157050453993, −7.03117849106627501533503077601, −6.52183409924032017598847199663, −5.84508098578725880263863151392, −5.37484584785383850457030776799, −3.99575772917284247763386451724, −3.86709904214332908635948878848, −2.66547321765012652301986475968, −1.60524385357388864055460274101, −1.05060246088758382568109167441,
1.05060246088758382568109167441, 1.60524385357388864055460274101, 2.66547321765012652301986475968, 3.86709904214332908635948878848, 3.99575772917284247763386451724, 5.37484584785383850457030776799, 5.84508098578725880263863151392, 6.52183409924032017598847199663, 7.03117849106627501533503077601, 7.998025692462453871157050453993
