| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 8-s − 2·9-s + 10-s − 11-s − 12-s − 2·13-s − 15-s + 16-s + 3·17-s − 2·18-s + 19-s + 20-s − 22-s + 6·23-s − 24-s + 25-s − 2·26-s + 5·27-s − 9·29-s − 30-s − 5·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.353·8-s − 2/3·9-s + 0.316·10-s − 0.301·11-s − 0.288·12-s − 0.554·13-s − 0.258·15-s + 1/4·16-s + 0.727·17-s − 0.471·18-s + 0.229·19-s + 0.223·20-s − 0.213·22-s + 1.25·23-s − 0.204·24-s + 1/5·25-s − 0.392·26-s + 0.962·27-s − 1.67·29-s − 0.182·30-s − 0.898·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.464861107\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.464861107\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−7.87129009845077588123512138414, −7.41935841988158226536953628236, −6.54393060077295121912679316228, −5.78833913191943630460797287603, −5.37510197912515163262683051708, −4.75910066016831688308666076963, −3.67859442360640402686306622375, −2.88973688056742863229922588523, −2.07095137810312987795444354413, −0.77089848495721519618058562539,
0.77089848495721519618058562539, 2.07095137810312987795444354413, 2.88973688056742863229922588523, 3.67859442360640402686306622375, 4.75910066016831688308666076963, 5.37510197912515163262683051708, 5.78833913191943630460797287603, 6.54393060077295121912679316228, 7.41935841988158226536953628236, 7.87129009845077588123512138414
