L(s) = 1 | − 2-s − 0.757·3-s + 4-s − 1.30·5-s + 0.757·6-s − 8-s − 2.42·9-s + 1.30·10-s + 1.45·11-s − 0.757·12-s − 3.28·13-s + 0.989·15-s + 16-s − 7.19·17-s + 2.42·18-s − 5.28·19-s − 1.30·20-s − 1.45·22-s − 3.28·23-s + 0.757·24-s − 3.29·25-s + 3.28·26-s + 4.10·27-s + 6.46·29-s − 0.989·30-s − 3.01·31-s − 32-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.437·3-s + 0.5·4-s − 0.584·5-s + 0.309·6-s − 0.353·8-s − 0.808·9-s + 0.413·10-s + 0.437·11-s − 0.218·12-s − 0.910·13-s + 0.255·15-s + 0.250·16-s − 1.74·17-s + 0.571·18-s − 1.21·19-s − 0.292·20-s − 0.309·22-s − 0.684·23-s + 0.154·24-s − 0.658·25-s + 0.643·26-s + 0.790·27-s + 1.19·29-s − 0.180·30-s − 0.540·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2971312214\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2971312214\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
good | 3 | \( 1 + 0.757T + 3T^{2} \) |
| 5 | \( 1 + 1.30T + 5T^{2} \) |
| 11 | \( 1 - 1.45T + 11T^{2} \) |
| 13 | \( 1 + 3.28T + 13T^{2} \) |
| 17 | \( 1 + 7.19T + 17T^{2} \) |
| 19 | \( 1 + 5.28T + 19T^{2} \) |
| 23 | \( 1 + 3.28T + 23T^{2} \) |
| 29 | \( 1 - 6.46T + 29T^{2} \) |
| 31 | \( 1 + 3.01T + 31T^{2} \) |
| 37 | \( 1 + 3.51T + 37T^{2} \) |
| 43 | \( 1 - 0.525T + 43T^{2} \) |
| 47 | \( 1 + 3.15T + 47T^{2} \) |
| 53 | \( 1 - 11.6T + 53T^{2} \) |
| 59 | \( 1 + 1.68T + 59T^{2} \) |
| 61 | \( 1 - 0.842T + 61T^{2} \) |
| 67 | \( 1 - 7.88T + 67T^{2} \) |
| 71 | \( 1 + 16.4T + 71T^{2} \) |
| 73 | \( 1 - 13.9T + 73T^{2} \) |
| 79 | \( 1 + 11.4T + 79T^{2} \) |
| 83 | \( 1 - 0.168T + 83T^{2} \) |
| 89 | \( 1 + 11.9T + 89T^{2} \) |
| 97 | \( 1 - 3.62T + 97T^{2} \) |
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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
−8.593795925745252983628941589844, −7.83297529628379447791154242557, −6.89752075000561427996918801577, −6.48064114537869426720774259984, −5.61304250300732347965019361747, −4.62032592422151729768338473786, −3.93469637650030537118840728836, −2.70172766854052546479075477054, −1.95890264744611950581604308450, −0.33251683220817613587042227718,
0.33251683220817613587042227718, 1.95890264744611950581604308450, 2.70172766854052546479075477054, 3.93469637650030537118840728836, 4.62032592422151729768338473786, 5.61304250300732347965019361747, 6.48064114537869426720774259984, 6.89752075000561427996918801577, 7.83297529628379447791154242557, 8.593795925745252983628941589844