| L(s) = 1 | − 3.41·5-s + 0.585·7-s + 2.82·11-s − 4.58·17-s − 2.24·19-s + 23-s + 6.65·25-s + 8.48·29-s + 8.48·31-s − 2·35-s + 0.828·37-s − 9.65·41-s + 10.2·43-s − 11.6·47-s − 6.65·49-s − 9.07·53-s − 9.65·55-s − 3.65·59-s + 4.82·61-s − 11.4·67-s − 2.34·71-s − 9.31·73-s + 1.65·77-s + 11.8·79-s + 1.17·83-s + 15.6·85-s + 1.07·89-s + ⋯ |
| L(s) = 1 | − 1.52·5-s + 0.221·7-s + 0.852·11-s − 1.11·17-s − 0.514·19-s + 0.208·23-s + 1.33·25-s + 1.57·29-s + 1.52·31-s − 0.338·35-s + 0.136·37-s − 1.50·41-s + 1.56·43-s − 1.70·47-s − 0.950·49-s − 1.24·53-s − 1.30·55-s − 0.476·59-s + 0.618·61-s − 1.39·67-s − 0.278·71-s − 1.09·73-s + 0.188·77-s + 1.33·79-s + 0.128·83-s + 1.69·85-s + 0.113·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + 3.41T + 5T^{2} \) |
| 7 | \( 1 - 0.585T + 7T^{2} \) |
| 11 | \( 1 - 2.82T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 4.58T + 17T^{2} \) |
| 19 | \( 1 + 2.24T + 19T^{2} \) |
| 29 | \( 1 - 8.48T + 29T^{2} \) |
| 31 | \( 1 - 8.48T + 31T^{2} \) |
| 37 | \( 1 - 0.828T + 37T^{2} \) |
| 41 | \( 1 + 9.65T + 41T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 47 | \( 1 + 11.6T + 47T^{2} \) |
| 53 | \( 1 + 9.07T + 53T^{2} \) |
| 59 | \( 1 + 3.65T + 59T^{2} \) |
| 61 | \( 1 - 4.82T + 61T^{2} \) |
| 67 | \( 1 + 11.4T + 67T^{2} \) |
| 71 | \( 1 + 2.34T + 71T^{2} \) |
| 73 | \( 1 + 9.31T + 73T^{2} \) |
| 79 | \( 1 - 11.8T + 79T^{2} \) |
| 83 | \( 1 - 1.17T + 83T^{2} \) |
| 89 | \( 1 - 1.07T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−8.321261589368194496601113101982, −7.62849113764404521516326650859, −6.66803283312637401925368436433, −6.35436529815680116660100980787, −4.75329775582676398279368611314, −4.50303588876900200951467634439, −3.59637235084828967407816971075, −2.71850969362066253889879738425, −1.31231782014517583598596464133, 0,
1.31231782014517583598596464133, 2.71850969362066253889879738425, 3.59637235084828967407816971075, 4.50303588876900200951467634439, 4.75329775582676398279368611314, 6.35436529815680116660100980787, 6.66803283312637401925368436433, 7.62849113764404521516326650859, 8.321261589368194496601113101982
