| L(s) = 1 | + 0.801·2-s − 1.80·3-s − 1.35·4-s − 2.44·5-s − 1.44·6-s − 7-s − 2.69·8-s + 0.246·9-s − 1.96·10-s + 1.49·11-s + 2.44·12-s + 2.93·13-s − 0.801·14-s + 4.40·15-s + 0.554·16-s − 5.93·17-s + 0.198·18-s − 3.02·19-s + 3.31·20-s + 1.80·21-s + 1.19·22-s + 4.74·23-s + 4.85·24-s + 0.978·25-s + 2.35·26-s + 4.96·27-s + 1.35·28-s + ⋯ |
| L(s) = 1 | + 0.567·2-s − 1.04·3-s − 0.678·4-s − 1.09·5-s − 0.589·6-s − 0.377·7-s − 0.951·8-s + 0.0823·9-s − 0.620·10-s + 0.450·11-s + 0.705·12-s + 0.815·13-s − 0.214·14-s + 1.13·15-s + 0.138·16-s − 1.44·17-s + 0.0466·18-s − 0.694·19-s + 0.741·20-s + 0.393·21-s + 0.255·22-s + 0.988·23-s + 0.990·24-s + 0.195·25-s + 0.462·26-s + 0.954·27-s + 0.256·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 151 | \( 1 + T \) |
| good | 2 | \( 1 - 0.801T + 2T^{2} \) |
| 3 | \( 1 + 1.80T + 3T^{2} \) |
| 5 | \( 1 + 2.44T + 5T^{2} \) |
| 7 | \( 1 + T + 7T^{2} \) |
| 11 | \( 1 - 1.49T + 11T^{2} \) |
| 13 | \( 1 - 2.93T + 13T^{2} \) |
| 17 | \( 1 + 5.93T + 17T^{2} \) |
| 19 | \( 1 + 3.02T + 19T^{2} \) |
| 23 | \( 1 - 4.74T + 23T^{2} \) |
| 29 | \( 1 + 9.25T + 29T^{2} \) |
| 31 | \( 1 + 5.18T + 31T^{2} \) |
| 37 | \( 1 - 1.06T + 37T^{2} \) |
| 41 | \( 1 + 0.902T + 41T^{2} \) |
| 43 | \( 1 - 8.82T + 43T^{2} \) |
| 47 | \( 1 + 11.1T + 47T^{2} \) |
| 53 | \( 1 + 5.26T + 53T^{2} \) |
| 59 | \( 1 - 1.91T + 59T^{2} \) |
| 61 | \( 1 - 7.00T + 61T^{2} \) |
| 67 | \( 1 - 0.241T + 67T^{2} \) |
| 71 | \( 1 - 7.64T + 71T^{2} \) |
| 73 | \( 1 + 2.82T + 73T^{2} \) |
| 79 | \( 1 + 3.37T + 79T^{2} \) |
| 83 | \( 1 + 4.38T + 83T^{2} \) |
| 89 | \( 1 + 12T + 89T^{2} \) |
| 97 | \( 1 - 5.07T + 97T^{2} \) |
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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
−12.64263757264192008343737549843, −11.36837301431931730265403324258, −11.08917759540738881127777253995, −9.321035124353913961411103214602, −8.412269380379253427330595278141, −6.81751110235975571105250679139, −5.80335896985344554077176299323, −4.56006782750996330048017116480, −3.59639775933490090524444896077, 0,
3.59639775933490090524444896077, 4.56006782750996330048017116480, 5.80335896985344554077176299323, 6.81751110235975571105250679139, 8.412269380379253427330595278141, 9.321035124353913961411103214602, 11.08917759540738881127777253995, 11.36837301431931730265403324258, 12.64263757264192008343737549843
