| L(s) = 1 | − 1.93·2-s − 1.04·3-s + 1.75·4-s − 5-s + 2.02·6-s − 2.21·7-s + 0.477·8-s − 1.90·9-s + 1.93·10-s − 0.841·11-s − 1.83·12-s + 4.89·13-s + 4.29·14-s + 1.04·15-s − 4.43·16-s + 3.85·17-s + 3.69·18-s − 0.266·19-s − 1.75·20-s + 2.31·21-s + 1.63·22-s − 2.96·23-s − 0.498·24-s + 25-s − 9.48·26-s + 5.12·27-s − 3.88·28-s + ⋯ |
| L(s) = 1 | − 1.36·2-s − 0.603·3-s + 0.876·4-s − 0.447·5-s + 0.826·6-s − 0.837·7-s + 0.168·8-s − 0.636·9-s + 0.612·10-s − 0.253·11-s − 0.528·12-s + 1.35·13-s + 1.14·14-s + 0.269·15-s − 1.10·16-s + 0.935·17-s + 0.871·18-s − 0.0610·19-s − 0.392·20-s + 0.505·21-s + 0.347·22-s − 0.618·23-s − 0.101·24-s + 0.200·25-s − 1.86·26-s + 0.986·27-s − 0.734·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 241 | \( 1 + T \) |
| good | 2 | \( 1 + 1.93T + 2T^{2} \) |
| 3 | \( 1 + 1.04T + 3T^{2} \) |
| 7 | \( 1 + 2.21T + 7T^{2} \) |
| 11 | \( 1 + 0.841T + 11T^{2} \) |
| 13 | \( 1 - 4.89T + 13T^{2} \) |
| 17 | \( 1 - 3.85T + 17T^{2} \) |
| 19 | \( 1 + 0.266T + 19T^{2} \) |
| 23 | \( 1 + 2.96T + 23T^{2} \) |
| 29 | \( 1 - 10.1T + 29T^{2} \) |
| 31 | \( 1 + 8.28T + 31T^{2} \) |
| 37 | \( 1 - 8.35T + 37T^{2} \) |
| 41 | \( 1 + 7.10T + 41T^{2} \) |
| 43 | \( 1 + 4.67T + 43T^{2} \) |
| 47 | \( 1 - 0.450T + 47T^{2} \) |
| 53 | \( 1 - 2.94T + 53T^{2} \) |
| 59 | \( 1 - 6.86T + 59T^{2} \) |
| 61 | \( 1 + 0.893T + 61T^{2} \) |
| 67 | \( 1 - 4.58T + 67T^{2} \) |
| 71 | \( 1 + 3.54T + 71T^{2} \) |
| 73 | \( 1 + 3.33T + 73T^{2} \) |
| 79 | \( 1 + 8.22T + 79T^{2} \) |
| 83 | \( 1 - 4.23T + 83T^{2} \) |
| 89 | \( 1 - 0.158T + 89T^{2} \) |
| 97 | \( 1 + 5.91T + 97T^{2} \) |
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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
−9.334478856539618033258809296651, −8.423982032388877820064240885084, −8.070427866891731046635114967824, −6.90587358849419762258947067672, −6.24033172935156737795660540812, −5.29340997611806954272776088314, −3.95314489354529732314057056870, −2.87045877026312118200877481583, −1.18691962160538669510004857734, 0,
1.18691962160538669510004857734, 2.87045877026312118200877481583, 3.95314489354529732314057056870, 5.29340997611806954272776088314, 6.24033172935156737795660540812, 6.90587358849419762258947067672, 8.070427866891731046635114967824, 8.423982032388877820064240885084, 9.334478856539618033258809296651
