| L(s) = 1 | − 2.49·2-s + 2·3-s + 4.20·4-s + 3.77·5-s − 4.98·6-s − 2·7-s − 5.49·8-s + 9-s − 9.40·10-s + 2.91·11-s + 8.40·12-s − 6.40·13-s + 4.98·14-s + 7.55·15-s + 5.26·16-s + 5.49·17-s − 2.49·18-s + 2.14·19-s + 15.8·20-s − 4·21-s − 7.26·22-s − 4.98·23-s − 10.9·24-s + 9.26·25-s + 15.9·26-s − 4·27-s − 8.40·28-s + ⋯ |
| L(s) = 1 | − 1.76·2-s + 1.15·3-s + 2.10·4-s + 1.68·5-s − 2.03·6-s − 0.755·7-s − 1.94·8-s + 0.333·9-s − 2.97·10-s + 0.879·11-s + 2.42·12-s − 1.77·13-s + 1.33·14-s + 1.95·15-s + 1.31·16-s + 1.33·17-s − 0.587·18-s + 0.491·19-s + 3.55·20-s − 0.872·21-s − 1.54·22-s − 1.03·23-s − 2.24·24-s + 1.85·25-s + 3.13·26-s − 0.769·27-s − 1.58·28-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
| \(L(1)\) |
\(\approx\) |
\(0.8600246652\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8600246652\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.49T + 2T^{2} \) |
| 3 | \( 1 - 2T + 3T^{2} \) |
| 5 | \( 1 - 3.77T + 5T^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 - 2.91T + 11T^{2} \) |
| 13 | \( 1 + 6.40T + 13T^{2} \) |
| 17 | \( 1 - 5.49T + 17T^{2} \) |
| 19 | \( 1 - 2.14T + 19T^{2} \) |
| 23 | \( 1 + 4.98T + 23T^{2} \) |
| 29 | \( 1 + 2.84T + 29T^{2} \) |
| 31 | \( 1 + 3.20T + 31T^{2} \) |
| 37 | \( 1 - 8.47T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 3.20T + 43T^{2} \) |
| 47 | \( 1 + 4.71T + 47T^{2} \) |
| 53 | \( 1 + 4.28T + 53T^{2} \) |
| 59 | \( 1 - 4.79T + 59T^{2} \) |
| 61 | \( 1 - 8.81T + 61T^{2} \) |
| 67 | \( 1 + 10.8T + 67T^{2} \) |
| 71 | \( 1 - 4.98T + 71T^{2} \) |
| 73 | \( 1 + 2.40T + 73T^{2} \) |
| 79 | \( 1 + 14.4T + 79T^{2} \) |
| 83 | \( 1 + 1.39T + 83T^{2} \) |
| 89 | \( 1 - 7.01T + 89T^{2} \) |
| 97 | \( 1 + 3.98T + 97T^{2} \) |
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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
−13.05302801578016212300011689190, −11.81193230865163819958309663868, −10.12000632852137901887870903661, −9.596360478784579741246088181297, −9.357105943568172504003842875613, −8.042365199299374944945461180041, −7.02709024980959392033863132334, −5.82778248384247945664751864632, −2.91890584097296150406560946324, −1.84665339994621814271803417974,
1.84665339994621814271803417974, 2.91890584097296150406560946324, 5.82778248384247945664751864632, 7.02709024980959392033863132334, 8.042365199299374944945461180041, 9.357105943568172504003842875613, 9.596360478784579741246088181297, 10.12000632852137901887870903661, 11.81193230865163819958309663868, 13.05302801578016212300011689190
