| L(s) = 1 | + 2·5-s + 11-s − 2·13-s + 6·17-s + 4·23-s − 25-s − 2·29-s − 10·37-s + 6·41-s + 8·43-s + 4·47-s + 6·53-s + 2·55-s + 12·59-s − 2·61-s − 4·65-s − 4·67-s + 12·71-s + 14·73-s − 16·79-s + 12·83-s + 12·85-s + 10·89-s + 14·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.301·11-s − 0.554·13-s + 1.45·17-s + 0.834·23-s − 1/5·25-s − 0.371·29-s − 1.64·37-s + 0.937·41-s + 1.21·43-s + 0.583·47-s + 0.824·53-s + 0.269·55-s + 1.56·59-s − 0.256·61-s − 0.496·65-s − 0.488·67-s + 1.42·71-s + 1.63·73-s − 1.80·79-s + 1.31·83-s + 1.30·85-s + 1.05·89-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.575623818\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.575623818\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−14.09838892743553, −13.63496001535028, −12.98642564776721, −12.54973284821797, −12.13638633740472, −11.57695990434959, −10.99984612620383, −10.33110911685657, −10.09950216082837, −9.472889403694006, −9.082715090810188, −8.581936058343933, −7.747392403588120, −7.431483342749265, −6.834693695979651, −6.204542670152558, −5.613392465451121, −5.304761322571862, −4.693343216845319, −3.790967066827057, −3.444821615905414, −2.518198315549080, −2.132242183174107, −1.267848042333298, −0.6606726009796458,
0.6606726009796458, 1.267848042333298, 2.132242183174107, 2.518198315549080, 3.444821615905414, 3.790967066827057, 4.693343216845319, 5.304761322571862, 5.613392465451121, 6.204542670152558, 6.834693695979651, 7.431483342749265, 7.747392403588120, 8.581936058343933, 9.082715090810188, 9.472889403694006, 10.09950216082837, 10.33110911685657, 10.99984612620383, 11.57695990434959, 12.13638633740472, 12.54973284821797, 12.98642564776721, 13.63496001535028, 14.09838892743553
