| L(s) = 1 | + 3-s − 1.23·5-s − 7-s + 9-s − 3.23·11-s + 13-s − 1.23·15-s − 2.47·17-s + 6.47·19-s − 21-s − 4.47·23-s − 3.47·25-s + 27-s + 8.47·29-s − 3.23·33-s + 1.23·35-s + 4.47·37-s + 39-s + 5.23·41-s − 4·43-s − 1.23·45-s − 7.70·47-s + 49-s − 2.47·51-s + 10·53-s + 4.00·55-s + 6.47·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.552·5-s − 0.377·7-s + 0.333·9-s − 0.975·11-s + 0.277·13-s − 0.319·15-s − 0.599·17-s + 1.48·19-s − 0.218·21-s − 0.932·23-s − 0.694·25-s + 0.192·27-s + 1.57·29-s − 0.563·33-s + 0.208·35-s + 0.735·37-s + 0.160·39-s + 0.817·41-s − 0.609·43-s − 0.184·45-s − 1.12·47-s + 0.142·49-s − 0.346·51-s + 1.37·53-s + 0.539·55-s + 0.857·57-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
| \(L(1)\) |
\(\approx\) |
\(1.777344517\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.777344517\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.23T + 5T^{2} \) |
| 11 | \( 1 + 3.23T + 11T^{2} \) |
| 17 | \( 1 + 2.47T + 17T^{2} \) |
| 19 | \( 1 - 6.47T + 19T^{2} \) |
| 23 | \( 1 + 4.47T + 23T^{2} \) |
| 29 | \( 1 - 8.47T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 4.47T + 37T^{2} \) |
| 41 | \( 1 - 5.23T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + 7.70T + 47T^{2} \) |
| 53 | \( 1 - 10T + 53T^{2} \) |
| 59 | \( 1 + 9.23T + 59T^{2} \) |
| 61 | \( 1 - 14.9T + 61T^{2} \) |
| 67 | \( 1 - 2.47T + 67T^{2} \) |
| 71 | \( 1 - 5.70T + 71T^{2} \) |
| 73 | \( 1 - 4.47T + 73T^{2} \) |
| 79 | \( 1 - 10.4T + 79T^{2} \) |
| 83 | \( 1 + 2.76T + 83T^{2} \) |
| 89 | \( 1 + 10.1T + 89T^{2} \) |
| 97 | \( 1 - 6.94T + 97T^{2} \) |
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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
−8.176078019252027014618335722467, −7.84242718317878053288558957976, −7.04764585010976774249601568199, −6.24501097147555027878919199697, −5.36439031719534470885979263544, −4.52486565273182306491484396779, −3.69827774714156860887465968385, −2.96571221190726358368805605801, −2.13513619651256155201511315243, −0.71368199335227499758098399258,
0.71368199335227499758098399258, 2.13513619651256155201511315243, 2.96571221190726358368805605801, 3.69827774714156860887465968385, 4.52486565273182306491484396779, 5.36439031719534470885979263544, 6.24501097147555027878919199697, 7.04764585010976774249601568199, 7.84242718317878053288558957976, 8.176078019252027014618335722467
