| L(s) = 1 | − 2-s + 4-s + 3·5-s − 8-s − 3·10-s + 3·11-s + 4·13-s + 16-s + 6·17-s + 7·19-s + 3·20-s − 3·22-s − 3·23-s + 4·25-s − 4·26-s − 5·31-s − 32-s − 6·34-s − 7·37-s − 7·38-s − 3·40-s + 9·41-s − 10·43-s + 3·44-s + 3·46-s − 6·47-s − 4·50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.34·5-s − 0.353·8-s − 0.948·10-s + 0.904·11-s + 1.10·13-s + 1/4·16-s + 1.45·17-s + 1.60·19-s + 0.670·20-s − 0.639·22-s − 0.625·23-s + 4/5·25-s − 0.784·26-s − 0.898·31-s − 0.176·32-s − 1.02·34-s − 1.15·37-s − 1.13·38-s − 0.474·40-s + 1.40·41-s − 1.52·43-s + 0.452·44-s + 0.442·46-s − 0.875·47-s − 0.565·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.093926342\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.093926342\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 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
−8.992251307828531296445343338023, −8.250819661564719776627888855362, −7.34475786918817202650312765182, −6.59247589213525769518809192749, −5.73304100710876992030851819199, −5.41376865336615198247524157398, −3.83254249011259881446516091298, −3.04507790741923758224766618955, −1.72414285314126353169175309195, −1.16041265383789495288315554440,
1.16041265383789495288315554440, 1.72414285314126353169175309195, 3.04507790741923758224766618955, 3.83254249011259881446516091298, 5.41376865336615198247524157398, 5.73304100710876992030851819199, 6.59247589213525769518809192749, 7.34475786918817202650312765182, 8.250819661564719776627888855362, 8.992251307828531296445343338023
