| L(s) = 1 | − 2-s + 3-s + 4-s − 2.38·5-s − 6-s + 0.127·7-s − 8-s + 9-s + 2.38·10-s + 11-s + 12-s + 5.55·13-s − 0.127·14-s − 2.38·15-s + 16-s + 7.68·17-s − 18-s − 2.88·19-s − 2.38·20-s + 0.127·21-s − 22-s − 3.73·23-s − 24-s + 0.676·25-s − 5.55·26-s + 27-s + 0.127·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.06·5-s − 0.408·6-s + 0.0482·7-s − 0.353·8-s + 0.333·9-s + 0.753·10-s + 0.301·11-s + 0.288·12-s + 1.53·13-s − 0.0341·14-s − 0.615·15-s + 0.250·16-s + 1.86·17-s − 0.235·18-s − 0.661·19-s − 0.532·20-s + 0.0278·21-s − 0.213·22-s − 0.778·23-s − 0.204·24-s + 0.135·25-s − 1.08·26-s + 0.192·27-s + 0.0241·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.531824961\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.531824961\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| good | 5 | \( 1 + 2.38T + 5T^{2} \) |
| 7 | \( 1 - 0.127T + 7T^{2} \) |
| 13 | \( 1 - 5.55T + 13T^{2} \) |
| 17 | \( 1 - 7.68T + 17T^{2} \) |
| 19 | \( 1 + 2.88T + 19T^{2} \) |
| 23 | \( 1 + 3.73T + 23T^{2} \) |
| 29 | \( 1 + 7.14T + 29T^{2} \) |
| 31 | \( 1 - 3.34T + 31T^{2} \) |
| 37 | \( 1 - 6.88T + 37T^{2} \) |
| 41 | \( 1 - 5.39T + 41T^{2} \) |
| 43 | \( 1 - 9.77T + 43T^{2} \) |
| 47 | \( 1 + 4.67T + 47T^{2} \) |
| 53 | \( 1 + 4.98T + 53T^{2} \) |
| 59 | \( 1 - 5.28T + 59T^{2} \) |
| 67 | \( 1 + 2.66T + 67T^{2} \) |
| 71 | \( 1 + 3.74T + 71T^{2} \) |
| 73 | \( 1 + 5.17T + 73T^{2} \) |
| 79 | \( 1 + 2.31T + 79T^{2} \) |
| 83 | \( 1 - 3.84T + 83T^{2} \) |
| 89 | \( 1 - 0.396T + 89T^{2} \) |
| 97 | \( 1 - 0.464T + 97T^{2} \) |
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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
−8.285154754377421916385168140194, −7.86590591465469253020689604733, −7.40083913951467898203901068481, −6.27287326676317750609627128213, −5.75717964856351567554566509396, −4.32508476157601068737417607177, −3.72920078987322313857671599229, −3.06245008037940278603170491294, −1.77415317025255983950731578988, −0.798959122639955287919416697658,
0.798959122639955287919416697658, 1.77415317025255983950731578988, 3.06245008037940278603170491294, 3.72920078987322313857671599229, 4.32508476157601068737417607177, 5.75717964856351567554566509396, 6.27287326676317750609627128213, 7.40083913951467898203901068481, 7.86590591465469253020689604733, 8.285154754377421916385168140194
