| L(s) = 1 | + 0.619·3-s − 3.40·5-s − 0.0449·7-s − 2.61·9-s − 3.92·11-s − 13-s − 2.11·15-s − 4.42·17-s − 7.70·19-s − 0.0278·21-s − 5.66·23-s + 6.62·25-s − 3.48·27-s + 29-s + 1.65·31-s − 2.43·33-s + 0.153·35-s + 3.38·37-s − 0.619·39-s − 0.932·41-s + 9.31·43-s + 8.91·45-s − 9.89·47-s − 6.99·49-s − 2.74·51-s − 5.95·53-s + 13.3·55-s + ⋯ |
| L(s) = 1 | + 0.357·3-s − 1.52·5-s − 0.0170·7-s − 0.871·9-s − 1.18·11-s − 0.277·13-s − 0.545·15-s − 1.07·17-s − 1.76·19-s − 0.00608·21-s − 1.18·23-s + 1.32·25-s − 0.669·27-s + 0.185·29-s + 0.296·31-s − 0.423·33-s + 0.0259·35-s + 0.556·37-s − 0.0992·39-s − 0.145·41-s + 1.41·43-s + 1.32·45-s − 1.44·47-s − 0.999·49-s − 0.383·51-s − 0.818·53-s + 1.80·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2211289783\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2211289783\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 - 0.619T + 3T^{2} \) |
| 5 | \( 1 + 3.40T + 5T^{2} \) |
| 7 | \( 1 + 0.0449T + 7T^{2} \) |
| 11 | \( 1 + 3.92T + 11T^{2} \) |
| 17 | \( 1 + 4.42T + 17T^{2} \) |
| 19 | \( 1 + 7.70T + 19T^{2} \) |
| 23 | \( 1 + 5.66T + 23T^{2} \) |
| 31 | \( 1 - 1.65T + 31T^{2} \) |
| 37 | \( 1 - 3.38T + 37T^{2} \) |
| 41 | \( 1 + 0.932T + 41T^{2} \) |
| 43 | \( 1 - 9.31T + 43T^{2} \) |
| 47 | \( 1 + 9.89T + 47T^{2} \) |
| 53 | \( 1 + 5.95T + 53T^{2} \) |
| 59 | \( 1 - 5.25T + 59T^{2} \) |
| 61 | \( 1 + 5.27T + 61T^{2} \) |
| 67 | \( 1 - 6.53T + 67T^{2} \) |
| 71 | \( 1 + 0.195T + 71T^{2} \) |
| 73 | \( 1 - 5.32T + 73T^{2} \) |
| 79 | \( 1 - 5.13T + 79T^{2} \) |
| 83 | \( 1 - 7.31T + 83T^{2} \) |
| 89 | \( 1 + 16.9T + 89T^{2} \) |
| 97 | \( 1 - 4.69T + 97T^{2} \) |
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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
−8.134303806896665368358816810583, −7.67891639029518448171842329897, −6.68692260947527702324339895808, −6.07089950008635958647662725764, −4.99007324372734765174186301573, −4.36392801356633310582177209278, −3.69559429666771843479505586961, −2.76061122196102400897058737789, −2.14670331183888610173501931548, −0.22191258497957777719583215318,
0.22191258497957777719583215318, 2.14670331183888610173501931548, 2.76061122196102400897058737789, 3.69559429666771843479505586961, 4.36392801356633310582177209278, 4.99007324372734765174186301573, 6.07089950008635958647662725764, 6.68692260947527702324339895808, 7.67891639029518448171842329897, 8.134303806896665368358816810583
