| L(s) = 1 | − 1.80·2-s + 1.24·3-s + 2.24·4-s − 2.24·6-s − 1.80·7-s − 2.24·8-s + 0.554·9-s + 2.80·12-s − 1.80·13-s + 3.24·14-s + 1.80·16-s − 0.999·18-s + 19-s − 2.24·21-s + 1.24·23-s − 2.80·24-s + 25-s + 3.24·26-s − 0.554·27-s − 4.04·28-s + 1.24·31-s − 1.00·32-s + 1.24·36-s − 1.80·38-s − 2.24·39-s + 4.04·42-s + 2·43-s + ⋯ |
| L(s) = 1 | − 1.80·2-s + 1.24·3-s + 2.24·4-s − 2.24·6-s − 1.80·7-s − 2.24·8-s + 0.554·9-s + 2.80·12-s − 1.80·13-s + 3.24·14-s + 1.80·16-s − 0.999·18-s + 19-s − 2.24·21-s + 1.24·23-s − 2.80·24-s + 25-s + 3.24·26-s − 0.554·27-s − 4.04·28-s + 1.24·31-s − 1.00·32-s + 1.24·36-s − 1.80·38-s − 2.24·39-s + 4.04·42-s + 2·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3743 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3743 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6162424159\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6162424159\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 - T \) |
| 197 | \( 1 - T \) |
| good | 2 | \( 1 + 1.80T + T^{2} \) |
| 3 | \( 1 - 1.24T + T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 + 1.80T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + 1.80T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 23 | \( 1 - 1.24T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - 1.24T + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - 2T + T^{2} \) |
| 47 | \( 1 + 1.80T + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - 1.24T + T^{2} \) |
| 67 | \( 1 - 1.24T + T^{2} \) |
| 71 | \( 1 + 0.445T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + 0.445T + T^{2} \) |
| 83 | \( 1 - 1.24T + T^{2} \) |
| 89 | \( 1 - 2T + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.007657863067461539061356917800, −8.073004496728418281471313753941, −7.37745679844401057446766421900, −6.97653319785175584070187676826, −6.22912453058522438492016890228, −4.96435582216199556814701850141, −3.44236753328410680101124634777, −2.79521078349710229975680959691, −2.38601551465650436388381342749, −0.792996553326606877913084380625,
0.792996553326606877913084380625, 2.38601551465650436388381342749, 2.79521078349710229975680959691, 3.44236753328410680101124634777, 4.96435582216199556814701850141, 6.22912453058522438492016890228, 6.97653319785175584070187676826, 7.37745679844401057446766421900, 8.073004496728418281471313753941, 9.007657863067461539061356917800
