| L(s) = 1 | + 2.29·2-s − 3-s + 3.25·4-s + 3.13·5-s − 2.29·6-s + 0.344·7-s + 2.86·8-s + 9-s + 7.18·10-s + 5.23·11-s − 3.25·12-s + 2.96·13-s + 0.789·14-s − 3.13·15-s + 0.0627·16-s − 17-s + 2.29·18-s − 2.69·19-s + 10.1·20-s − 0.344·21-s + 11.9·22-s + 7.52·23-s − 2.86·24-s + 4.83·25-s + 6.79·26-s − 27-s + 1.11·28-s + ⋯ |
| L(s) = 1 | + 1.62·2-s − 0.577·3-s + 1.62·4-s + 1.40·5-s − 0.935·6-s + 0.130·7-s + 1.01·8-s + 0.333·9-s + 2.27·10-s + 1.57·11-s − 0.938·12-s + 0.822·13-s + 0.210·14-s − 0.809·15-s + 0.0156·16-s − 0.242·17-s + 0.540·18-s − 0.618·19-s + 2.27·20-s − 0.0751·21-s + 2.55·22-s + 1.56·23-s − 0.584·24-s + 0.967·25-s + 1.33·26-s − 0.192·27-s + 0.211·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.155026200\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.155026200\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 79 | \( 1 - T \) |
| good | 2 | \( 1 - 2.29T + 2T^{2} \) |
| 5 | \( 1 - 3.13T + 5T^{2} \) |
| 7 | \( 1 - 0.344T + 7T^{2} \) |
| 11 | \( 1 - 5.23T + 11T^{2} \) |
| 13 | \( 1 - 2.96T + 13T^{2} \) |
| 19 | \( 1 + 2.69T + 19T^{2} \) |
| 23 | \( 1 - 7.52T + 23T^{2} \) |
| 29 | \( 1 + 1.77T + 29T^{2} \) |
| 31 | \( 1 + 0.447T + 31T^{2} \) |
| 37 | \( 1 + 0.229T + 37T^{2} \) |
| 41 | \( 1 + 7.81T + 41T^{2} \) |
| 43 | \( 1 + 6.20T + 43T^{2} \) |
| 47 | \( 1 + 3.09T + 47T^{2} \) |
| 53 | \( 1 - 1.46T + 53T^{2} \) |
| 59 | \( 1 - 10.7T + 59T^{2} \) |
| 61 | \( 1 - 8.00T + 61T^{2} \) |
| 67 | \( 1 + 10.2T + 67T^{2} \) |
| 71 | \( 1 + 0.297T + 71T^{2} \) |
| 73 | \( 1 + 5.70T + 73T^{2} \) |
| 83 | \( 1 - 4.51T + 83T^{2} \) |
| 89 | \( 1 - 13.7T + 89T^{2} \) |
| 97 | \( 1 - 12.0T + 97T^{2} \) |
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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
−8.628377904274578813129201795738, −7.08734714880179482851263522986, −6.47477593055870477848570183705, −6.24746723934719662331075032142, −5.36840280932526333243320652525, −4.86029738508107795003595226851, −3.95681500953710041479356956020, −3.23036731785201653146022887994, −2.04580437997307290003746077004, −1.33247273421014261575785677066,
1.33247273421014261575785677066, 2.04580437997307290003746077004, 3.23036731785201653146022887994, 3.95681500953710041479356956020, 4.86029738508107795003595226851, 5.36840280932526333243320652525, 6.24746723934719662331075032142, 6.47477593055870477848570183705, 7.08734714880179482851263522986, 8.628377904274578813129201795738
