| L(s) = 1 | − 6.90·3-s + 21.0·5-s − 7·7-s + 20.7·9-s − 26.7·11-s − 22.6·13-s − 145.·15-s − 17·17-s − 42.0·19-s + 48.3·21-s + 36.7·23-s + 319.·25-s + 43.3·27-s + 130.·29-s − 14.9·31-s + 184.·33-s − 147.·35-s − 294.·37-s + 156.·39-s − 257.·41-s − 403.·43-s + 437.·45-s − 375.·47-s + 49·49-s + 117.·51-s + 87.9·53-s − 563.·55-s + ⋯ |
| L(s) = 1 | − 1.32·3-s + 1.88·5-s − 0.377·7-s + 0.767·9-s − 0.732·11-s − 0.482·13-s − 2.50·15-s − 0.242·17-s − 0.508·19-s + 0.502·21-s + 0.333·23-s + 2.55·25-s + 0.308·27-s + 0.836·29-s − 0.0868·31-s + 0.974·33-s − 0.712·35-s − 1.30·37-s + 0.641·39-s − 0.982·41-s − 1.43·43-s + 1.44·45-s − 1.16·47-s + 0.142·49-s + 0.322·51-s + 0.227·53-s − 1.38·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| 17 | \( 1 + 17T \) |
| good | 3 | \( 1 + 6.90T + 27T^{2} \) |
| 5 | \( 1 - 21.0T + 125T^{2} \) |
| 11 | \( 1 + 26.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 22.6T + 2.19e3T^{2} \) |
| 19 | \( 1 + 42.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 36.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 130.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 14.9T + 2.97e4T^{2} \) |
| 37 | \( 1 + 294.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 257.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 403.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 375.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 87.9T + 1.48e5T^{2} \) |
| 59 | \( 1 + 355.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 485.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 583.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.00e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.19e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 335.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 92.1T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.22e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 821.T + 9.12e5T^{2} \) |
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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
−10.30663377857558371950949560607, −9.577726615089389285596481936484, −8.500845102932199369732339317792, −6.81982533250410075416645551896, −6.36735892490720936002300456700, −5.36867584358959672240917851767, −4.92293615459741022199003020002, −2.83757196308899321541137841803, −1.59936235608862876180399519149, 0,
1.59936235608862876180399519149, 2.83757196308899321541137841803, 4.92293615459741022199003020002, 5.36867584358959672240917851767, 6.36735892490720936002300456700, 6.81982533250410075416645551896, 8.500845102932199369732339317792, 9.577726615089389285596481936484, 10.30663377857558371950949560607
