| L(s) = 1 | + 1.83·2-s − 2.76·3-s + 1.37·4-s − 5.08·6-s − 4.65·7-s − 1.15·8-s + 4.67·9-s + 3.19·11-s − 3.80·12-s − 8.54·14-s − 4.86·16-s + 4.71·17-s + 8.57·18-s + 6.40·19-s + 12.8·21-s + 5.87·22-s − 1.22·23-s + 3.18·24-s − 4.62·27-s − 6.38·28-s + 5.29·29-s − 5.50·31-s − 6.62·32-s − 8.85·33-s + 8.65·34-s + 6.41·36-s + 4.08·37-s + ⋯ |
| L(s) = 1 | + 1.29·2-s − 1.59·3-s + 0.686·4-s − 2.07·6-s − 1.75·7-s − 0.406·8-s + 1.55·9-s + 0.963·11-s − 1.09·12-s − 2.28·14-s − 1.21·16-s + 1.14·17-s + 2.02·18-s + 1.47·19-s + 2.81·21-s + 1.25·22-s − 0.255·23-s + 0.650·24-s − 0.890·27-s − 1.20·28-s + 0.983·29-s − 0.988·31-s − 1.17·32-s − 1.54·33-s + 1.48·34-s + 1.06·36-s + 0.672·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 1.83T + 2T^{2} \) |
| 3 | \( 1 + 2.76T + 3T^{2} \) |
| 7 | \( 1 + 4.65T + 7T^{2} \) |
| 11 | \( 1 - 3.19T + 11T^{2} \) |
| 17 | \( 1 - 4.71T + 17T^{2} \) |
| 19 | \( 1 - 6.40T + 19T^{2} \) |
| 23 | \( 1 + 1.22T + 23T^{2} \) |
| 29 | \( 1 - 5.29T + 29T^{2} \) |
| 31 | \( 1 + 5.50T + 31T^{2} \) |
| 37 | \( 1 - 4.08T + 37T^{2} \) |
| 41 | \( 1 + 1.78T + 41T^{2} \) |
| 43 | \( 1 - 4.57T + 43T^{2} \) |
| 47 | \( 1 + 10.9T + 47T^{2} \) |
| 53 | \( 1 + 7.13T + 53T^{2} \) |
| 59 | \( 1 + 4.07T + 59T^{2} \) |
| 61 | \( 1 + 10.7T + 61T^{2} \) |
| 67 | \( 1 + 11.8T + 67T^{2} \) |
| 71 | \( 1 - 3.95T + 71T^{2} \) |
| 73 | \( 1 - 4.85T + 73T^{2} \) |
| 79 | \( 1 + 4.26T + 79T^{2} \) |
| 83 | \( 1 + 13.8T + 83T^{2} \) |
| 89 | \( 1 + 0.0287T + 89T^{2} \) |
| 97 | \( 1 - 9.96T + 97T^{2} \) |
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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
−7.56140177164235074996066118117, −6.80871784486997922387481821382, −6.20488710998484973203141309940, −5.89408400530305841354745492664, −5.15948532305499554782084781953, −4.38617222754953611294806775245, −3.48137561967260615439912582740, −3.02861339469115987805884717225, −1.19769775789386342318210668956, 0,
1.19769775789386342318210668956, 3.02861339469115987805884717225, 3.48137561967260615439912582740, 4.38617222754953611294806775245, 5.15948532305499554782084781953, 5.89408400530305841354745492664, 6.20488710998484973203141309940, 6.80871784486997922387481821382, 7.56140177164235074996066118117
