| L(s) = 1 | − 2.61·2-s + 4.81·4-s − 3.81·5-s − 7.34·8-s + 9.95·10-s + 4.73·11-s − 13-s + 9.55·16-s − 5.22·17-s − 2.92·19-s − 18.3·20-s − 12.3·22-s − 3.33·23-s + 9.55·25-s + 2.61·26-s + 0.922·29-s + 7.51·31-s − 10.2·32-s + 13.6·34-s + 0.154·37-s + 7.62·38-s + 28.0·40-s + 6.36·41-s − 6.55·43-s + 22.8·44-s + 8.69·46-s + 9.03·47-s + ⋯ |
| L(s) = 1 | − 1.84·2-s + 2.40·4-s − 1.70·5-s − 2.59·8-s + 3.14·10-s + 1.42·11-s − 0.277·13-s + 2.38·16-s − 1.26·17-s − 0.670·19-s − 4.10·20-s − 2.63·22-s − 0.694·23-s + 1.91·25-s + 0.511·26-s + 0.171·29-s + 1.35·31-s − 1.81·32-s + 2.33·34-s + 0.0254·37-s + 1.23·38-s + 4.43·40-s + 0.994·41-s − 0.999·43-s + 3.43·44-s + 1.28·46-s + 1.31·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5733 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5733 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 + 2.61T + 2T^{2} \) |
| 5 | \( 1 + 3.81T + 5T^{2} \) |
| 11 | \( 1 - 4.73T + 11T^{2} \) |
| 17 | \( 1 + 5.22T + 17T^{2} \) |
| 19 | \( 1 + 2.92T + 19T^{2} \) |
| 23 | \( 1 + 3.33T + 23T^{2} \) |
| 29 | \( 1 - 0.922T + 29T^{2} \) |
| 31 | \( 1 - 7.51T + 31T^{2} \) |
| 37 | \( 1 - 0.154T + 37T^{2} \) |
| 41 | \( 1 - 6.36T + 41T^{2} \) |
| 43 | \( 1 + 6.55T + 43T^{2} \) |
| 47 | \( 1 - 9.03T + 47T^{2} \) |
| 53 | \( 1 + 8.55T + 53T^{2} \) |
| 59 | \( 1 - 3.95T + 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 + 10.6T + 67T^{2} \) |
| 71 | \( 1 - 6.58T + 71T^{2} \) |
| 73 | \( 1 - 7.73T + 73T^{2} \) |
| 79 | \( 1 - 13.3T + 79T^{2} \) |
| 83 | \( 1 + 1.40T + 83T^{2} \) |
| 89 | \( 1 + 1.96T + 89T^{2} \) |
| 97 | \( 1 - 2.11T + 97T^{2} \) |
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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
−8.018418909771570376364735413933, −7.28129006344996689288496539812, −6.65051742935430940916553814124, −6.22990041440798402899152382151, −4.56729295247629563909694239013, −4.02809745942750714628333507919, −3.01951775836285878222624563232, −2.02128732798285343792710446009, −0.917038887603706134983942892517, 0,
0.917038887603706134983942892517, 2.02128732798285343792710446009, 3.01951775836285878222624563232, 4.02809745942750714628333507919, 4.56729295247629563909694239013, 6.22990041440798402899152382151, 6.65051742935430940916553814124, 7.28129006344996689288496539812, 8.018418909771570376364735413933
