| L(s) = 1 | + 2-s − 0.425·3-s + 4-s + 2.71·5-s − 0.425·6-s − 0.709·7-s + 8-s − 2.81·9-s + 2.71·10-s + 4.35·11-s − 0.425·12-s − 3.12·13-s − 0.709·14-s − 1.15·15-s + 16-s + 2.44·17-s − 2.81·18-s + 1.64·19-s + 2.71·20-s + 0.301·21-s + 4.35·22-s + 23-s − 0.425·24-s + 2.38·25-s − 3.12·26-s + 2.47·27-s − 0.709·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.245·3-s + 0.5·4-s + 1.21·5-s − 0.173·6-s − 0.268·7-s + 0.353·8-s − 0.939·9-s + 0.859·10-s + 1.31·11-s − 0.122·12-s − 0.865·13-s − 0.189·14-s − 0.298·15-s + 0.250·16-s + 0.594·17-s − 0.664·18-s + 0.377·19-s + 0.607·20-s + 0.0658·21-s + 0.928·22-s + 0.208·23-s − 0.0868·24-s + 0.476·25-s − 0.611·26-s + 0.476·27-s − 0.134·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.734011936\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.734011936\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 131 | \( 1 - T \) |
| good | 3 | \( 1 + 0.425T + 3T^{2} \) |
| 5 | \( 1 - 2.71T + 5T^{2} \) |
| 7 | \( 1 + 0.709T + 7T^{2} \) |
| 11 | \( 1 - 4.35T + 11T^{2} \) |
| 13 | \( 1 + 3.12T + 13T^{2} \) |
| 17 | \( 1 - 2.44T + 17T^{2} \) |
| 19 | \( 1 - 1.64T + 19T^{2} \) |
| 29 | \( 1 + 1.66T + 29T^{2} \) |
| 31 | \( 1 - 8.07T + 31T^{2} \) |
| 37 | \( 1 - 2.79T + 37T^{2} \) |
| 41 | \( 1 + 2.91T + 41T^{2} \) |
| 43 | \( 1 + 0.195T + 43T^{2} \) |
| 47 | \( 1 - 7.43T + 47T^{2} \) |
| 53 | \( 1 - 7.60T + 53T^{2} \) |
| 59 | \( 1 - 4.52T + 59T^{2} \) |
| 61 | \( 1 + 14.6T + 61T^{2} \) |
| 67 | \( 1 + 5.81T + 67T^{2} \) |
| 71 | \( 1 - 7.57T + 71T^{2} \) |
| 73 | \( 1 + 1.23T + 73T^{2} \) |
| 79 | \( 1 - 15.5T + 79T^{2} \) |
| 83 | \( 1 - 0.456T + 83T^{2} \) |
| 89 | \( 1 - 9.41T + 89T^{2} \) |
| 97 | \( 1 - 13.4T + 97T^{2} \) |
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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
−7.979338156216233510703883712665, −7.11509009650277853083335056816, −6.33166766414162760492012571638, −6.01812755866032558971525497910, −5.27350106587921891086181347951, −4.62717761802945078072776067156, −3.57593146332814915299642949256, −2.81371366941968328561968020115, −2.03061929809865358591775965061, −0.948501090835917052768207831778,
0.948501090835917052768207831778, 2.03061929809865358591775965061, 2.81371366941968328561968020115, 3.57593146332814915299642949256, 4.62717761802945078072776067156, 5.27350106587921891086181347951, 6.01812755866032558971525497910, 6.33166766414162760492012571638, 7.11509009650277853083335056816, 7.979338156216233510703883712665
