L(s) = 1 | − 3-s + 5-s + 5·7-s + 9-s − 3·11-s − 15-s − 17-s + 7·19-s − 5·21-s − 6·23-s + 25-s − 27-s + 9·29-s − 6·31-s + 3·33-s + 5·35-s + 9·37-s − 3·41-s − 6·43-s + 45-s − 9·47-s + 18·49-s + 51-s + 3·53-s − 3·55-s − 7·57-s + 14·59-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1.88·7-s + 1/3·9-s − 0.904·11-s − 0.258·15-s − 0.242·17-s + 1.60·19-s − 1.09·21-s − 1.25·23-s + 1/5·25-s − 0.192·27-s + 1.67·29-s − 1.07·31-s + 0.522·33-s + 0.845·35-s + 1.47·37-s − 0.468·41-s − 0.914·43-s + 0.149·45-s − 1.31·47-s + 18/7·49-s + 0.140·51-s + 0.412·53-s − 0.404·55-s − 0.927·57-s + 1.82·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 344760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 344760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.600316118\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.600316118\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| 17 | \( 1 + T \) |
good | 7 | \( 1 - 5 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 9 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−12.58319553033756, −11.86830609317857, −11.63096679162888, −11.30336584901465, −10.82410958261957, −10.31536038280204, −9.918336689746741, −9.551997150912615, −8.842448656268757, −8.166687650571060, −8.037665262933213, −7.699215736587294, −6.858773633322453, −6.666700486145507, −5.813350612901152, −5.379541196158304, −5.125835790464960, −4.754349136817391, −4.096850809379740, −3.567554597199031, −2.698481797250125, −2.241484644412868, −1.708546185867027, −1.099466486473567, −0.5599534017638584,
0.5599534017638584, 1.099466486473567, 1.708546185867027, 2.241484644412868, 2.698481797250125, 3.567554597199031, 4.096850809379740, 4.754349136817391, 5.125835790464960, 5.379541196158304, 5.813350612901152, 6.666700486145507, 6.858773633322453, 7.699215736587294, 8.037665262933213, 8.166687650571060, 8.842448656268757, 9.551997150912615, 9.918336689746741, 10.31536038280204, 10.82410958261957, 11.30336584901465, 11.63096679162888, 11.86830609317857, 12.58319553033756