| L(s) = 1 | − 1.78·3-s + 0.474·5-s + 7-s + 0.177·9-s + 11-s + 13-s − 0.846·15-s + 0.796·17-s + 5.88·19-s − 1.78·21-s + 2.43·23-s − 4.77·25-s + 5.03·27-s + 4.63·29-s − 3.56·31-s − 1.78·33-s + 0.474·35-s − 3.78·37-s − 1.78·39-s + 4.10·41-s + 7.57·43-s + 0.0842·45-s − 3.82·47-s + 49-s − 1.41·51-s + 8.74·53-s + 0.474·55-s + ⋯ |
| L(s) = 1 | − 1.02·3-s + 0.212·5-s + 0.377·7-s + 0.0591·9-s + 0.301·11-s + 0.277·13-s − 0.218·15-s + 0.193·17-s + 1.35·19-s − 0.388·21-s + 0.507·23-s − 0.954·25-s + 0.968·27-s + 0.859·29-s − 0.640·31-s − 0.310·33-s + 0.0802·35-s − 0.622·37-s − 0.285·39-s + 0.640·41-s + 1.15·43-s + 0.0125·45-s − 0.557·47-s + 0.142·49-s − 0.198·51-s + 1.20·53-s + 0.0640·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.527463106\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.527463106\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + 1.78T + 3T^{2} \) |
| 5 | \( 1 - 0.474T + 5T^{2} \) |
| 17 | \( 1 - 0.796T + 17T^{2} \) |
| 19 | \( 1 - 5.88T + 19T^{2} \) |
| 23 | \( 1 - 2.43T + 23T^{2} \) |
| 29 | \( 1 - 4.63T + 29T^{2} \) |
| 31 | \( 1 + 3.56T + 31T^{2} \) |
| 37 | \( 1 + 3.78T + 37T^{2} \) |
| 41 | \( 1 - 4.10T + 41T^{2} \) |
| 43 | \( 1 - 7.57T + 43T^{2} \) |
| 47 | \( 1 + 3.82T + 47T^{2} \) |
| 53 | \( 1 - 8.74T + 53T^{2} \) |
| 59 | \( 1 + 12.4T + 59T^{2} \) |
| 61 | \( 1 - 3.42T + 61T^{2} \) |
| 67 | \( 1 - 5.90T + 67T^{2} \) |
| 71 | \( 1 + 2.54T + 71T^{2} \) |
| 73 | \( 1 + 0.114T + 73T^{2} \) |
| 79 | \( 1 + 2.34T + 79T^{2} \) |
| 83 | \( 1 + 13.4T + 83T^{2} \) |
| 89 | \( 1 - 15.2T + 89T^{2} \) |
| 97 | \( 1 + 4.55T + 97T^{2} \) |
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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
−7.68918071439059133029301199720, −7.10662774713510239194726764764, −6.29633009445428506583165232102, −5.70071494266599210085167695398, −5.22383911631082964724426984549, −4.46576194531792074129013337741, −3.56205382359630960856187722331, −2.68052656444936438026072394594, −1.52420223770823624436863815559, −0.68890201918395893563111644347,
0.68890201918395893563111644347, 1.52420223770823624436863815559, 2.68052656444936438026072394594, 3.56205382359630960856187722331, 4.46576194531792074129013337741, 5.22383911631082964724426984549, 5.70071494266599210085167695398, 6.29633009445428506583165232102, 7.10662774713510239194726764764, 7.68918071439059133029301199720
