| L(s) = 1 | + 3-s − 5-s + 2.37·7-s + 9-s − 2.37·11-s − 13-s − 15-s + 4.37·17-s − 4.74·19-s + 2.37·21-s + 6.37·23-s + 25-s + 27-s − 2·29-s + 4.74·31-s − 2.37·33-s − 2.37·35-s − 0.372·37-s − 39-s + 4.37·41-s + 4·43-s − 45-s + 12.7·47-s − 1.37·49-s + 4.37·51-s − 3.62·53-s + 2.37·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.896·7-s + 0.333·9-s − 0.715·11-s − 0.277·13-s − 0.258·15-s + 1.06·17-s − 1.08·19-s + 0.517·21-s + 1.32·23-s + 0.200·25-s + 0.192·27-s − 0.371·29-s + 0.852·31-s − 0.412·33-s − 0.400·35-s − 0.0612·37-s − 0.160·39-s + 0.682·41-s + 0.609·43-s − 0.149·45-s + 1.85·47-s − 0.196·49-s + 0.612·51-s − 0.498·53-s + 0.319·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.330363036\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.330363036\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| good | 7 | \( 1 - 2.37T + 7T^{2} \) |
| 11 | \( 1 + 2.37T + 11T^{2} \) |
| 17 | \( 1 - 4.37T + 17T^{2} \) |
| 19 | \( 1 + 4.74T + 19T^{2} \) |
| 23 | \( 1 - 6.37T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 4.74T + 31T^{2} \) |
| 37 | \( 1 + 0.372T + 37T^{2} \) |
| 41 | \( 1 - 4.37T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 - 12.7T + 47T^{2} \) |
| 53 | \( 1 + 3.62T + 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 + 9.11T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 - 5.62T + 71T^{2} \) |
| 73 | \( 1 - 7.48T + 73T^{2} \) |
| 79 | \( 1 + 11.8T + 79T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 - 7.62T + 89T^{2} \) |
| 97 | \( 1 - 8.37T + 97T^{2} \) |
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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
−8.624584731082655362424883369530, −7.81505967036070242013497267022, −7.56604460644555693041563239528, −6.55583370893388076493927889733, −5.47464369187667541921420252408, −4.78101814352305884474049274050, −4.00645512161330821042273752053, −2.99446207764179010787073874349, −2.18110608425180761642306814475, −0.923718055399996828974956536915,
0.923718055399996828974956536915, 2.18110608425180761642306814475, 2.99446207764179010787073874349, 4.00645512161330821042273752053, 4.78101814352305884474049274050, 5.47464369187667541921420252408, 6.55583370893388076493927889733, 7.56604460644555693041563239528, 7.81505967036070242013497267022, 8.624584731082655362424883369530
