| L(s) = 1 | − 0.656·3-s + 0.656·7-s − 2.56·9-s + 0.343·11-s + 1.91·13-s − 4.48·17-s + 19-s − 0.431·21-s − 3.56·23-s + 3.65·27-s + 7.99·29-s − 5.73·31-s − 0.225·33-s + 4.16·37-s − 1.25·39-s − 9.08·41-s + 3.51·43-s + 3.40·47-s − 6.56·49-s + 2.94·51-s + 1.68·53-s − 0.656·57-s + 7.82·59-s − 12.4·61-s − 1.68·63-s + 9.48·67-s + 2.34·69-s + ⋯ |
| L(s) = 1 | − 0.379·3-s + 0.248·7-s − 0.856·9-s + 0.103·11-s + 0.530·13-s − 1.08·17-s + 0.229·19-s − 0.0940·21-s − 0.744·23-s + 0.703·27-s + 1.48·29-s − 1.03·31-s − 0.0392·33-s + 0.685·37-s − 0.201·39-s − 1.41·41-s + 0.535·43-s + 0.496·47-s − 0.938·49-s + 0.412·51-s + 0.231·53-s − 0.0869·57-s + 1.01·59-s − 1.59·61-s − 0.212·63-s + 1.15·67-s + 0.282·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.335790061\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.335790061\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 0.656T + 3T^{2} \) |
| 7 | \( 1 - 0.656T + 7T^{2} \) |
| 11 | \( 1 - 0.343T + 11T^{2} \) |
| 13 | \( 1 - 1.91T + 13T^{2} \) |
| 17 | \( 1 + 4.48T + 17T^{2} \) |
| 23 | \( 1 + 3.56T + 23T^{2} \) |
| 29 | \( 1 - 7.99T + 29T^{2} \) |
| 31 | \( 1 + 5.73T + 31T^{2} \) |
| 37 | \( 1 - 4.16T + 37T^{2} \) |
| 41 | \( 1 + 9.08T + 41T^{2} \) |
| 43 | \( 1 - 3.51T + 43T^{2} \) |
| 47 | \( 1 - 3.40T + 47T^{2} \) |
| 53 | \( 1 - 1.68T + 53T^{2} \) |
| 59 | \( 1 - 7.82T + 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 - 9.48T + 67T^{2} \) |
| 71 | \( 1 - 9.96T + 71T^{2} \) |
| 73 | \( 1 + 7.53T + 73T^{2} \) |
| 79 | \( 1 - 14.9T + 79T^{2} \) |
| 83 | \( 1 + 10.9T + 83T^{2} \) |
| 89 | \( 1 + 4.19T + 89T^{2} \) |
| 97 | \( 1 + 1.73T + 97T^{2} \) |
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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
−8.056370416250078149241609146757, −7.04359078618064592789397688194, −6.43251347559225452320834435285, −5.84284584137753179560479485599, −5.09743560431841196345257280817, −4.40235447793232113066141594491, −3.54846785922016636985988336956, −2.67546180104255472066642430831, −1.78615308264847434476178639419, −0.58162228428375042775719492280,
0.58162228428375042775719492280, 1.78615308264847434476178639419, 2.67546180104255472066642430831, 3.54846785922016636985988336956, 4.40235447793232113066141594491, 5.09743560431841196345257280817, 5.84284584137753179560479485599, 6.43251347559225452320834435285, 7.04359078618064592789397688194, 8.056370416250078149241609146757
