| L(s) = 1 | − 2.24·2-s + 1.24·3-s + 3.04·4-s − 3.80·5-s − 2.80·6-s − 7-s − 2.35·8-s − 1.44·9-s + 8.54·10-s − 1.89·11-s + 3.80·12-s + 0.911·13-s + 2.24·14-s − 4.74·15-s − 0.801·16-s − 3.91·17-s + 3.24·18-s − 6.76·19-s − 11.5·20-s − 1.24·21-s + 4.24·22-s − 0.335·23-s − 2.93·24-s + 9.45·25-s − 2.04·26-s − 5.54·27-s − 3.04·28-s + ⋯ |
| L(s) = 1 | − 1.58·2-s + 0.719·3-s + 1.52·4-s − 1.70·5-s − 1.14·6-s − 0.377·7-s − 0.833·8-s − 0.481·9-s + 2.70·10-s − 0.569·11-s + 1.09·12-s + 0.252·13-s + 0.600·14-s − 1.22·15-s − 0.200·16-s − 0.948·17-s + 0.765·18-s − 1.55·19-s − 2.59·20-s − 0.272·21-s + 0.905·22-s − 0.0698·23-s − 0.599·24-s + 1.89·25-s − 0.401·26-s − 1.06·27-s − 0.576·28-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 | \( 1 + 2.24T + 2T^{2} \) |
| 3 | \( 1 - 1.24T + 3T^{2} \) |
| 5 | \( 1 + 3.80T + 5T^{2} \) |
| 7 | \( 1 + T + 7T^{2} \) |
| 11 | \( 1 + 1.89T + 11T^{2} \) |
| 13 | \( 1 - 0.911T + 13T^{2} \) |
| 17 | \( 1 + 3.91T + 17T^{2} \) |
| 19 | \( 1 + 6.76T + 19T^{2} \) |
| 23 | \( 1 + 0.335T + 23T^{2} \) |
| 29 | \( 1 - 7.67T + 29T^{2} \) |
| 31 | \( 1 + 1.46T + 31T^{2} \) |
| 37 | \( 1 - 3.08T + 37T^{2} \) |
| 41 | \( 1 + 10.3T + 41T^{2} \) |
| 43 | \( 1 - 9.51T + 43T^{2} \) |
| 47 | \( 1 - 7.76T + 47T^{2} \) |
| 53 | \( 1 - 4.89T + 53T^{2} \) |
| 59 | \( 1 - 8.36T + 59T^{2} \) |
| 61 | \( 1 + 8.23T + 61T^{2} \) |
| 67 | \( 1 + 13.6T + 67T^{2} \) |
| 71 | \( 1 + 9.96T + 71T^{2} \) |
| 73 | \( 1 - 8.70T + 73T^{2} \) |
| 79 | \( 1 - 9.50T + 79T^{2} \) |
| 83 | \( 1 + 3.71T + 83T^{2} \) |
| 89 | \( 1 + 12T + 89T^{2} \) |
| 97 | \( 1 - 4.07T + 97T^{2} \) |
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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
−12.17216837269419025187999821068, −11.14734889176512569971637913916, −10.45629811529634858964459711111, −8.963339421493218907695425541432, −8.443379859838028181513847224563, −7.73456457059618442474646554773, −6.64119553276885441127218537456, −4.21740718778131784248779329532, −2.67245042091945515097468418581, 0,
2.67245042091945515097468418581, 4.21740718778131784248779329532, 6.64119553276885441127218537456, 7.73456457059618442474646554773, 8.443379859838028181513847224563, 8.963339421493218907695425541432, 10.45629811529634858964459711111, 11.14734889176512569971637913916, 12.17216837269419025187999821068
