| L(s) = 1 | − 2-s − 3.09·3-s + 4-s + 2.78·5-s + 3.09·6-s − 8-s + 6.59·9-s − 2.78·10-s + 2.80·11-s − 3.09·12-s + 3.76·13-s − 8.63·15-s + 16-s − 6.21·17-s − 6.59·18-s + 1.46·19-s + 2.78·20-s − 2.80·22-s − 6.75·23-s + 3.09·24-s + 2.78·25-s − 3.76·26-s − 11.1·27-s − 29-s + 8.63·30-s − 8.37·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.78·3-s + 0.5·4-s + 1.24·5-s + 1.26·6-s − 0.353·8-s + 2.19·9-s − 0.882·10-s + 0.845·11-s − 0.894·12-s + 1.04·13-s − 2.23·15-s + 0.250·16-s − 1.50·17-s − 1.55·18-s + 0.336·19-s + 0.623·20-s − 0.597·22-s − 1.40·23-s + 0.632·24-s + 0.556·25-s − 0.738·26-s − 2.14·27-s − 0.185·29-s + 1.57·30-s − 1.50·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 + 3.09T + 3T^{2} \) |
| 5 | \( 1 - 2.78T + 5T^{2} \) |
| 11 | \( 1 - 2.80T + 11T^{2} \) |
| 13 | \( 1 - 3.76T + 13T^{2} \) |
| 17 | \( 1 + 6.21T + 17T^{2} \) |
| 19 | \( 1 - 1.46T + 19T^{2} \) |
| 23 | \( 1 + 6.75T + 23T^{2} \) |
| 31 | \( 1 + 8.37T + 31T^{2} \) |
| 37 | \( 1 - 2.28T + 37T^{2} \) |
| 41 | \( 1 + 8.55T + 41T^{2} \) |
| 43 | \( 1 - 7.71T + 43T^{2} \) |
| 47 | \( 1 + 12.2T + 47T^{2} \) |
| 53 | \( 1 + 1.12T + 53T^{2} \) |
| 59 | \( 1 + 3.38T + 59T^{2} \) |
| 61 | \( 1 + 3.70T + 61T^{2} \) |
| 67 | \( 1 + 6.57T + 67T^{2} \) |
| 71 | \( 1 - 8.45T + 71T^{2} \) |
| 73 | \( 1 - 1.41T + 73T^{2} \) |
| 79 | \( 1 + 12.8T + 79T^{2} \) |
| 83 | \( 1 + 7.72T + 83T^{2} \) |
| 89 | \( 1 - 9.99T + 89T^{2} \) |
| 97 | \( 1 + 1.13T + 97T^{2} \) |
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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
−8.631509128708268912849941632655, −7.42327166255827033084293746886, −6.55786588097311149716633518680, −6.19308515284106321364083122687, −5.68853265383860220304860501761, −4.72154612788047022961399845983, −3.74436519499949213051488585556, −1.97598899702177952756366671634, −1.39066000625654046556418209050, 0,
1.39066000625654046556418209050, 1.97598899702177952756366671634, 3.74436519499949213051488585556, 4.72154612788047022961399845983, 5.68853265383860220304860501761, 6.19308515284106321364083122687, 6.55786588097311149716633518680, 7.42327166255827033084293746886, 8.631509128708268912849941632655
