| L(s) = 1 | − 0.126·3-s − 4.03·5-s − 7-s − 2.98·9-s − 3.52·11-s + 2.41·13-s + 0.511·15-s − 0.0534·17-s + 4.51·19-s + 0.126·21-s + 2.50·23-s + 11.3·25-s + 0.757·27-s + 8.96·29-s + 0.562·31-s + 0.446·33-s + 4.03·35-s − 3.34·37-s − 0.305·39-s − 41-s + 2.96·43-s + 12.0·45-s − 6.57·47-s + 49-s + 0.00677·51-s − 0.414·53-s + 14.2·55-s + ⋯ |
| L(s) = 1 | − 0.0731·3-s − 1.80·5-s − 0.377·7-s − 0.994·9-s − 1.06·11-s + 0.669·13-s + 0.132·15-s − 0.0129·17-s + 1.03·19-s + 0.0276·21-s + 0.522·23-s + 2.26·25-s + 0.145·27-s + 1.66·29-s + 0.101·31-s + 0.0777·33-s + 0.682·35-s − 0.549·37-s − 0.0489·39-s − 0.156·41-s + 0.452·43-s + 1.79·45-s − 0.958·47-s + 0.142·49-s + 0.000948·51-s − 0.0569·53-s + 1.91·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7548344441\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7548344441\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 41 | \( 1 + T \) |
| good | 3 | \( 1 + 0.126T + 3T^{2} \) |
| 5 | \( 1 + 4.03T + 5T^{2} \) |
| 11 | \( 1 + 3.52T + 11T^{2} \) |
| 13 | \( 1 - 2.41T + 13T^{2} \) |
| 17 | \( 1 + 0.0534T + 17T^{2} \) |
| 19 | \( 1 - 4.51T + 19T^{2} \) |
| 23 | \( 1 - 2.50T + 23T^{2} \) |
| 29 | \( 1 - 8.96T + 29T^{2} \) |
| 31 | \( 1 - 0.562T + 31T^{2} \) |
| 37 | \( 1 + 3.34T + 37T^{2} \) |
| 43 | \( 1 - 2.96T + 43T^{2} \) |
| 47 | \( 1 + 6.57T + 47T^{2} \) |
| 53 | \( 1 + 0.414T + 53T^{2} \) |
| 59 | \( 1 + 10.6T + 59T^{2} \) |
| 61 | \( 1 - 6.49T + 61T^{2} \) |
| 67 | \( 1 - 12.1T + 67T^{2} \) |
| 71 | \( 1 - 10.4T + 71T^{2} \) |
| 73 | \( 1 + 3.94T + 73T^{2} \) |
| 79 | \( 1 + 9.62T + 79T^{2} \) |
| 83 | \( 1 - 6.61T + 83T^{2} \) |
| 89 | \( 1 + 13.8T + 89T^{2} \) |
| 97 | \( 1 + 11.8T + 97T^{2} \) |
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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
−9.852027710565908344496623244606, −8.619066023350237601264500561071, −8.256355297154572026960770737727, −7.45161558498026496029827075703, −6.59407524249801155281972542580, −5.43350430043639101107768774734, −4.59767555820586536513878101642, −3.41998826947386337090939018701, −2.90111465176379692916680174203, −0.63568383522265433275784968898,
0.63568383522265433275784968898, 2.90111465176379692916680174203, 3.41998826947386337090939018701, 4.59767555820586536513878101642, 5.43350430043639101107768774734, 6.59407524249801155281972542580, 7.45161558498026496029827075703, 8.256355297154572026960770737727, 8.619066023350237601264500561071, 9.852027710565908344496623244606
