| L(s) = 1 | − 1.21·2-s − 0.688·3-s − 0.525·4-s − 5-s + 0.836·6-s + 7-s + 3.06·8-s − 2.52·9-s + 1.21·10-s + 0.361·12-s + 3.73·13-s − 1.21·14-s + 0.688·15-s − 2.67·16-s − 0.0666·17-s + 3.06·18-s − 6.42·19-s + 0.525·20-s − 0.688·21-s − 1.09·23-s − 2.11·24-s + 25-s − 4.54·26-s + 3.80·27-s − 0.525·28-s + 7.80·29-s − 0.836·30-s + ⋯ |
| L(s) = 1 | − 0.858·2-s − 0.397·3-s − 0.262·4-s − 0.447·5-s + 0.341·6-s + 0.377·7-s + 1.08·8-s − 0.841·9-s + 0.384·10-s + 0.104·12-s + 1.03·13-s − 0.324·14-s + 0.177·15-s − 0.668·16-s − 0.0161·17-s + 0.722·18-s − 1.47·19-s + 0.117·20-s − 0.150·21-s − 0.228·23-s − 0.431·24-s + 0.200·25-s − 0.890·26-s + 0.732·27-s − 0.0992·28-s + 1.44·29-s − 0.152·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 1.21T + 2T^{2} \) |
| 3 | \( 1 + 0.688T + 3T^{2} \) |
| 13 | \( 1 - 3.73T + 13T^{2} \) |
| 17 | \( 1 + 0.0666T + 17T^{2} \) |
| 19 | \( 1 + 6.42T + 19T^{2} \) |
| 23 | \( 1 + 1.09T + 23T^{2} \) |
| 29 | \( 1 - 7.80T + 29T^{2} \) |
| 31 | \( 1 + 5.59T + 31T^{2} \) |
| 37 | \( 1 - 1.33T + 37T^{2} \) |
| 41 | \( 1 + 6.64T + 41T^{2} \) |
| 43 | \( 1 - 11.7T + 43T^{2} \) |
| 47 | \( 1 + 2.26T + 47T^{2} \) |
| 53 | \( 1 + 1.71T + 53T^{2} \) |
| 59 | \( 1 - 2.54T + 59T^{2} \) |
| 61 | \( 1 + 14.4T + 61T^{2} \) |
| 67 | \( 1 - 10.3T + 67T^{2} \) |
| 71 | \( 1 + 12.5T + 71T^{2} \) |
| 73 | \( 1 + 1.17T + 73T^{2} \) |
| 79 | \( 1 - 8.51T + 79T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 - 15.6T + 89T^{2} \) |
| 97 | \( 1 + 13.4T + 97T^{2} \) |
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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
−8.225598530648414696092947857332, −7.58865313164770722728634366841, −6.57099044643292256822477397115, −5.95727993395610976695065466334, −4.96865993925329474311816956920, −4.32280712671900465423079312014, −3.45966285607318574460234225772, −2.20105613770269576807152234021, −1.05285008971948718870510312235, 0,
1.05285008971948718870510312235, 2.20105613770269576807152234021, 3.45966285607318574460234225772, 4.32280712671900465423079312014, 4.96865993925329474311816956920, 5.95727993395610976695065466334, 6.57099044643292256822477397115, 7.58865313164770722728634366841, 8.225598530648414696092947857332
