| L(s) = 1 | − 3-s − 5-s − 7-s + 9-s − 5·11-s − 13-s + 15-s − 3·17-s + 19-s + 21-s − 3·23-s − 4·25-s − 27-s + 9·29-s − 4·31-s + 5·33-s + 35-s − 11·37-s + 39-s + 5·43-s − 45-s + 8·47-s + 49-s + 3·51-s − 2·53-s + 5·55-s − 57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 1.50·11-s − 0.277·13-s + 0.258·15-s − 0.727·17-s + 0.229·19-s + 0.218·21-s − 0.625·23-s − 4/5·25-s − 0.192·27-s + 1.67·29-s − 0.718·31-s + 0.870·33-s + 0.169·35-s − 1.80·37-s + 0.160·39-s + 0.762·43-s − 0.149·45-s + 1.16·47-s + 1/7·49-s + 0.420·51-s − 0.274·53-s + 0.674·55-s − 0.132·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6121405627\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6121405627\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 15 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−8.201446606753346861375494180077, −7.64539992819326394416926284918, −6.93384459729236425490988967620, −6.14870499157140837975820876453, −5.37589820595753831680253807256, −4.73726907840251180750454379627, −3.87503086378985783714020107268, −2.90477588733476870765553764796, −1.99947627461773626388751061444, −0.42907642254029688922261495011,
0.42907642254029688922261495011, 1.99947627461773626388751061444, 2.90477588733476870765553764796, 3.87503086378985783714020107268, 4.73726907840251180750454379627, 5.37589820595753831680253807256, 6.14870499157140837975820876453, 6.93384459729236425490988967620, 7.64539992819326394416926284918, 8.201446606753346861375494180077
