| L(s) = 1 | + 1.96·3-s + 1.54·7-s + 0.849·9-s − 4.56·11-s − 1.09·13-s + 17-s − 4.67·19-s + 3.03·21-s − 0.529·23-s − 4.21·27-s + 8.06·29-s + 4.78·31-s − 8.94·33-s − 5.27·37-s − 2.15·39-s − 0.751·41-s − 9.49·43-s − 10.7·47-s − 4.61·49-s + 1.96·51-s + 0.0227·53-s − 9.17·57-s + 3.56·59-s + 3.92·61-s + 1.31·63-s + 9.75·67-s − 1.03·69-s + ⋯ |
| L(s) = 1 | + 1.13·3-s + 0.583·7-s + 0.283·9-s − 1.37·11-s − 0.304·13-s + 0.242·17-s − 1.07·19-s + 0.661·21-s − 0.110·23-s − 0.812·27-s + 1.49·29-s + 0.858·31-s − 1.55·33-s − 0.867·37-s − 0.344·39-s − 0.117·41-s − 1.44·43-s − 1.56·47-s − 0.659·49-s + 0.274·51-s + 0.00313·53-s − 1.21·57-s + 0.464·59-s + 0.502·61-s + 0.165·63-s + 1.19·67-s − 0.125·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 - 1.96T + 3T^{2} \) |
| 7 | \( 1 - 1.54T + 7T^{2} \) |
| 11 | \( 1 + 4.56T + 11T^{2} \) |
| 13 | \( 1 + 1.09T + 13T^{2} \) |
| 19 | \( 1 + 4.67T + 19T^{2} \) |
| 23 | \( 1 + 0.529T + 23T^{2} \) |
| 29 | \( 1 - 8.06T + 29T^{2} \) |
| 31 | \( 1 - 4.78T + 31T^{2} \) |
| 37 | \( 1 + 5.27T + 37T^{2} \) |
| 41 | \( 1 + 0.751T + 41T^{2} \) |
| 43 | \( 1 + 9.49T + 43T^{2} \) |
| 47 | \( 1 + 10.7T + 47T^{2} \) |
| 53 | \( 1 - 0.0227T + 53T^{2} \) |
| 59 | \( 1 - 3.56T + 59T^{2} \) |
| 61 | \( 1 - 3.92T + 61T^{2} \) |
| 67 | \( 1 - 9.75T + 67T^{2} \) |
| 71 | \( 1 + 1.21T + 71T^{2} \) |
| 73 | \( 1 - 10.1T + 73T^{2} \) |
| 79 | \( 1 + 14.4T + 79T^{2} \) |
| 83 | \( 1 - 5.08T + 83T^{2} \) |
| 89 | \( 1 + 17.6T + 89T^{2} \) |
| 97 | \( 1 + 6.78T + 97T^{2} \) |
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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
−7.966900405145238733475139003161, −7.04445295642574824002639546645, −6.34948045278581884908489656353, −5.25533103918527424272330506124, −4.82959886260957063944878767803, −3.87179594566174400710332652523, −2.95769664289084469231084976957, −2.46414169923693007483064779010, −1.58410490228433161795675349580, 0,
1.58410490228433161795675349580, 2.46414169923693007483064779010, 2.95769664289084469231084976957, 3.87179594566174400710332652523, 4.82959886260957063944878767803, 5.25533103918527424272330506124, 6.34948045278581884908489656353, 7.04445295642574824002639546645, 7.966900405145238733475139003161
