| L(s) = 1 | + 1.28·5-s − 2.78·7-s + 1.78·13-s + 4.86·17-s − 5.35·19-s + 7.57·23-s − 3.35·25-s − 5.28·29-s − 0.223·31-s − 3.57·35-s − 3.56·37-s − 6.29·41-s − 2.56·43-s + 4·47-s + 0.776·49-s − 2.71·53-s − 0.422·59-s − 1.77·61-s + 2.29·65-s − 6.78·67-s − 8.70·71-s + 12.9·73-s − 6.93·79-s − 9.13·83-s + 6.23·85-s + 16.8·89-s − 4.98·91-s + ⋯ |
| L(s) = 1 | + 0.573·5-s − 1.05·7-s + 0.496·13-s + 1.17·17-s − 1.22·19-s + 1.58·23-s − 0.670·25-s − 0.980·29-s − 0.0400·31-s − 0.604·35-s − 0.586·37-s − 0.983·41-s − 0.391·43-s + 0.583·47-s + 0.110·49-s − 0.373·53-s − 0.0550·59-s − 0.227·61-s + 0.284·65-s − 0.829·67-s − 1.03·71-s + 1.51·73-s − 0.779·79-s − 1.00·83-s + 0.676·85-s + 1.78·89-s − 0.522·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 - 1.28T + 5T^{2} \) |
| 7 | \( 1 + 2.78T + 7T^{2} \) |
| 13 | \( 1 - 1.78T + 13T^{2} \) |
| 17 | \( 1 - 4.86T + 17T^{2} \) |
| 19 | \( 1 + 5.35T + 19T^{2} \) |
| 23 | \( 1 - 7.57T + 23T^{2} \) |
| 29 | \( 1 + 5.28T + 29T^{2} \) |
| 31 | \( 1 + 0.223T + 31T^{2} \) |
| 37 | \( 1 + 3.56T + 37T^{2} \) |
| 41 | \( 1 + 6.29T + 41T^{2} \) |
| 43 | \( 1 + 2.56T + 43T^{2} \) |
| 47 | \( 1 - 4T + 47T^{2} \) |
| 53 | \( 1 + 2.71T + 53T^{2} \) |
| 59 | \( 1 + 0.422T + 59T^{2} \) |
| 61 | \( 1 + 1.77T + 61T^{2} \) |
| 67 | \( 1 + 6.78T + 67T^{2} \) |
| 71 | \( 1 + 8.70T + 71T^{2} \) |
| 73 | \( 1 - 12.9T + 73T^{2} \) |
| 79 | \( 1 + 6.93T + 79T^{2} \) |
| 83 | \( 1 + 9.13T + 83T^{2} \) |
| 89 | \( 1 - 16.8T + 89T^{2} \) |
| 97 | \( 1 - 17.2T + 97T^{2} \) |
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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
−7.31353353239757660076252884649, −6.66510435821977154917024321165, −6.05554000926272874667874757322, −5.50869162595489382852408394875, −4.67782200943385814448866876534, −3.62346767495751200724483784831, −3.21754031797840626207404183227, −2.19961004045164720890123562152, −1.28300176963049452047610548112, 0,
1.28300176963049452047610548112, 2.19961004045164720890123562152, 3.21754031797840626207404183227, 3.62346767495751200724483784831, 4.67782200943385814448866876534, 5.50869162595489382852408394875, 6.05554000926272874667874757322, 6.66510435821977154917024321165, 7.31353353239757660076252884649
