| L(s) = 1 | + 2-s − 2.45·3-s + 4-s − 1.39·5-s − 2.45·6-s + 7-s + 8-s + 3.00·9-s − 1.39·10-s − 4.53·11-s − 2.45·12-s − 4.61·13-s + 14-s + 3.41·15-s + 16-s − 6.23·17-s + 3.00·18-s − 2.76·19-s − 1.39·20-s − 2.45·21-s − 4.53·22-s − 2.68·23-s − 2.45·24-s − 3.06·25-s − 4.61·26-s − 0.0126·27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.41·3-s + 0.5·4-s − 0.622·5-s − 1.00·6-s + 0.377·7-s + 0.353·8-s + 1.00·9-s − 0.440·10-s − 1.36·11-s − 0.707·12-s − 1.28·13-s + 0.267·14-s + 0.880·15-s + 0.250·16-s − 1.51·17-s + 0.708·18-s − 0.634·19-s − 0.311·20-s − 0.534·21-s − 0.967·22-s − 0.559·23-s − 0.500·24-s − 0.612·25-s − 0.905·26-s − 0.00244·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5032354215\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5032354215\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 431 | \( 1 - T \) |
| good | 3 | \( 1 + 2.45T + 3T^{2} \) |
| 5 | \( 1 + 1.39T + 5T^{2} \) |
| 11 | \( 1 + 4.53T + 11T^{2} \) |
| 13 | \( 1 + 4.61T + 13T^{2} \) |
| 17 | \( 1 + 6.23T + 17T^{2} \) |
| 19 | \( 1 + 2.76T + 19T^{2} \) |
| 23 | \( 1 + 2.68T + 23T^{2} \) |
| 29 | \( 1 - 2.77T + 29T^{2} \) |
| 31 | \( 1 - 1.77T + 31T^{2} \) |
| 37 | \( 1 + 1.21T + 37T^{2} \) |
| 41 | \( 1 - 1.19T + 41T^{2} \) |
| 43 | \( 1 + 7.87T + 43T^{2} \) |
| 47 | \( 1 + 8.76T + 47T^{2} \) |
| 53 | \( 1 - 12.2T + 53T^{2} \) |
| 59 | \( 1 - 2.27T + 59T^{2} \) |
| 61 | \( 1 - 7.57T + 61T^{2} \) |
| 67 | \( 1 - 11.9T + 67T^{2} \) |
| 71 | \( 1 - 1.41T + 71T^{2} \) |
| 73 | \( 1 + 11.4T + 73T^{2} \) |
| 79 | \( 1 - 4.66T + 79T^{2} \) |
| 83 | \( 1 + 11.4T + 83T^{2} \) |
| 89 | \( 1 - 14.1T + 89T^{2} \) |
| 97 | \( 1 + 16.3T + 97T^{2} \) |
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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
−7.989874274565435751214160568830, −7.01975671027070704844733464563, −6.69138189412940931663257591657, −5.72352180656483295302099306964, −5.15093235428665493098444849183, −4.63580925583356528897432042540, −4.03472170933434789287131862596, −2.72282680525920740697203094531, −2.01023577823348532720220293984, −0.33506846283663985979950483986,
0.33506846283663985979950483986, 2.01023577823348532720220293984, 2.72282680525920740697203094531, 4.03472170933434789287131862596, 4.63580925583356528897432042540, 5.15093235428665493098444849183, 5.72352180656483295302099306964, 6.69138189412940931663257591657, 7.01975671027070704844733464563, 7.989874274565435751214160568830
