| L(s) = 1 | + 2-s − 2.60·3-s + 4-s + 0.145·5-s − 2.60·6-s + 0.982·7-s + 8-s + 3.80·9-s + 0.145·10-s − 4.17·11-s − 2.60·12-s + 2.84·13-s + 0.982·14-s − 0.379·15-s + 16-s − 1.12·17-s + 3.80·18-s − 4.48·19-s + 0.145·20-s − 2.56·21-s − 4.17·22-s + 6.95·23-s − 2.60·24-s − 4.97·25-s + 2.84·26-s − 2.09·27-s + 0.982·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.50·3-s + 0.5·4-s + 0.0651·5-s − 1.06·6-s + 0.371·7-s + 0.353·8-s + 1.26·9-s + 0.0460·10-s − 1.26·11-s − 0.753·12-s + 0.788·13-s + 0.262·14-s − 0.0980·15-s + 0.250·16-s − 0.272·17-s + 0.896·18-s − 1.02·19-s + 0.0325·20-s − 0.559·21-s − 0.890·22-s + 1.45·23-s − 0.532·24-s − 0.995·25-s + 0.557·26-s − 0.403·27-s + 0.185·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 4021 | \( 1+O(T) \) |
| good | 3 | \( 1 + 2.60T + 3T^{2} \) |
| 5 | \( 1 - 0.145T + 5T^{2} \) |
| 7 | \( 1 - 0.982T + 7T^{2} \) |
| 11 | \( 1 + 4.17T + 11T^{2} \) |
| 13 | \( 1 - 2.84T + 13T^{2} \) |
| 17 | \( 1 + 1.12T + 17T^{2} \) |
| 19 | \( 1 + 4.48T + 19T^{2} \) |
| 23 | \( 1 - 6.95T + 23T^{2} \) |
| 29 | \( 1 + 7.09T + 29T^{2} \) |
| 31 | \( 1 - 5.43T + 31T^{2} \) |
| 37 | \( 1 - 11.1T + 37T^{2} \) |
| 41 | \( 1 + 0.660T + 41T^{2} \) |
| 43 | \( 1 - 7.34T + 43T^{2} \) |
| 47 | \( 1 + 9.13T + 47T^{2} \) |
| 53 | \( 1 + 2.95T + 53T^{2} \) |
| 59 | \( 1 - 14.3T + 59T^{2} \) |
| 61 | \( 1 + 9.35T + 61T^{2} \) |
| 67 | \( 1 + 3.06T + 67T^{2} \) |
| 71 | \( 1 + 6.29T + 71T^{2} \) |
| 73 | \( 1 + 13.1T + 73T^{2} \) |
| 79 | \( 1 + 7.99T + 79T^{2} \) |
| 83 | \( 1 + 5.16T + 83T^{2} \) |
| 89 | \( 1 - 11.6T + 89T^{2} \) |
| 97 | \( 1 - 8.46T + 97T^{2} \) |
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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
−7.32893745916828710279250429745, −6.45748910584159228541791217469, −5.97427277542267278135280176785, −5.47864838742349871741764113389, −4.69279956059242155193164507476, −4.32350718393292333587311520540, −3.17466568632612351969483176091, −2.23870357663079064995344815239, −1.18430344000879347809092558049, 0,
1.18430344000879347809092558049, 2.23870357663079064995344815239, 3.17466568632612351969483176091, 4.32350718393292333587311520540, 4.69279956059242155193164507476, 5.47864838742349871741764113389, 5.97427277542267278135280176785, 6.45748910584159228541791217469, 7.32893745916828710279250429745
