| L(s) = 1 | − 2-s + 4-s − 8-s − 5·11-s − 13-s + 16-s − 8·19-s + 5·22-s − 4·23-s + 26-s − 5·29-s − 32-s − 2·37-s + 8·38-s + 9·41-s − 3·43-s − 5·44-s + 4·46-s + 8·47-s − 52-s + 5·53-s + 5·58-s + 13·59-s + 2·61-s + 64-s + 8·67-s − 2·71-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 1.50·11-s − 0.277·13-s + 1/4·16-s − 1.83·19-s + 1.06·22-s − 0.834·23-s + 0.196·26-s − 0.928·29-s − 0.176·32-s − 0.328·37-s + 1.29·38-s + 1.40·41-s − 0.457·43-s − 0.753·44-s + 0.589·46-s + 1.16·47-s − 0.138·52-s + 0.686·53-s + 0.656·58-s + 1.69·59-s + 0.256·61-s + 1/8·64-s + 0.977·67-s − 0.237·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6372944655\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6372944655\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 11 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 - 13 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−12.68307645124223, −12.32771480004869, −11.76093065737877, −11.13908456649646, −10.85331320862074, −10.34703396419574, −10.11368662684314, −9.520607397632445, −8.987374696389612, −8.528249475856734, −8.075467535463303, −7.749192218497898, −7.174833269585851, −6.780392574400920, −6.020854174111072, −5.813279355007601, −5.121088153821635, −4.679292975764793, −3.873225759795656, −3.630480015165140, −2.553302311978833, −2.340303396105883, −1.972639152006907, −0.9307427136579523, −0.2742200503402086,
0.2742200503402086, 0.9307427136579523, 1.972639152006907, 2.340303396105883, 2.553302311978833, 3.630480015165140, 3.873225759795656, 4.679292975764793, 5.121088153821635, 5.813279355007601, 6.020854174111072, 6.780392574400920, 7.174833269585851, 7.749192218497898, 8.075467535463303, 8.528249475856734, 8.987374696389612, 9.520607397632445, 10.11368662684314, 10.34703396419574, 10.85331320862074, 11.13908456649646, 11.76093065737877, 12.32771480004869, 12.68307645124223
