| L(s) = 1 | + 1.14e14·2-s + 8.75e22·3-s − 2.65e28·4-s − 1.07e33·5-s + 1.00e37·6-s + 1.86e40·7-s − 7.56e42·8-s + 5.55e45·9-s − 1.23e47·10-s + 1.86e49·11-s − 2.32e51·12-s − 5.82e52·13-s + 2.13e54·14-s − 9.43e55·15-s + 1.85e56·16-s − 1.12e58·17-s + 6.35e59·18-s + 8.30e60·19-s + 2.85e61·20-s + 1.63e63·21-s + 2.13e63·22-s + 1.09e64·23-s − 6.62e65·24-s − 1.36e66·25-s − 6.66e66·26-s + 3.00e68·27-s − 4.93e68·28-s + ⋯ |
| L(s) = 1 | + 0.574·2-s + 1.90·3-s − 0.669·4-s − 0.678·5-s + 1.09·6-s + 1.34·7-s − 0.959·8-s + 2.61·9-s − 0.389·10-s + 0.638·11-s − 1.27·12-s − 0.713·13-s + 0.771·14-s − 1.28·15-s + 0.117·16-s − 0.403·17-s + 1.50·18-s + 1.50·19-s + 0.454·20-s + 2.55·21-s + 0.367·22-s + 0.227·23-s − 1.82·24-s − 0.540·25-s − 0.409·26-s + 3.07·27-s − 0.898·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(96-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+95/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(48)\) |
\(\approx\) |
\(5.243455489\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.243455489\) |
| \(L(\frac{97}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 - 1.14e14T + 3.96e28T^{2} \) |
| 3 | \( 1 - 8.75e22T + 2.12e45T^{2} \) |
| 5 | \( 1 + 1.07e33T + 2.52e66T^{2} \) |
| 7 | \( 1 - 1.86e40T + 1.92e80T^{2} \) |
| 11 | \( 1 - 1.86e49T + 8.55e98T^{2} \) |
| 13 | \( 1 + 5.82e52T + 6.67e105T^{2} \) |
| 17 | \( 1 + 1.12e58T + 7.80e116T^{2} \) |
| 19 | \( 1 - 8.30e60T + 3.03e121T^{2} \) |
| 23 | \( 1 - 1.09e64T + 2.31e129T^{2} \) |
| 29 | \( 1 - 2.28e69T + 8.46e138T^{2} \) |
| 31 | \( 1 - 3.05e70T + 4.77e141T^{2} \) |
| 37 | \( 1 - 3.99e74T + 9.53e148T^{2} \) |
| 41 | \( 1 + 3.81e76T + 1.63e153T^{2} \) |
| 43 | \( 1 - 2.50e77T + 1.51e155T^{2} \) |
| 47 | \( 1 - 2.28e79T + 7.06e158T^{2} \) |
| 53 | \( 1 + 5.87e80T + 6.40e163T^{2} \) |
| 59 | \( 1 - 1.48e84T + 1.70e168T^{2} \) |
| 61 | \( 1 + 5.46e84T + 4.03e169T^{2} \) |
| 67 | \( 1 + 6.62e86T + 2.99e173T^{2} \) |
| 71 | \( 1 - 5.25e87T + 7.40e175T^{2} \) |
| 73 | \( 1 + 2.09e88T + 1.03e177T^{2} \) |
| 79 | \( 1 - 6.42e89T + 1.88e180T^{2} \) |
| 83 | \( 1 - 1.28e91T + 2.05e182T^{2} \) |
| 89 | \( 1 + 5.79e92T + 1.55e185T^{2} \) |
| 97 | \( 1 + 1.99e94T + 5.53e188T^{2} \) |
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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
−14.44650495679029834746295060204, −13.65506776472535092069837079094, −11.98835428651433940881288822793, −9.557293839991645149513140061386, −8.436233611732001849032754425952, −7.51419797134169025823416805375, −4.74810634005191650756778103560, −3.90257631357927310604650880200, −2.69063185476504745146116031101, −1.18103627470192005905805839381,
1.18103627470192005905805839381, 2.69063185476504745146116031101, 3.90257631357927310604650880200, 4.74810634005191650756778103560, 7.51419797134169025823416805375, 8.436233611732001849032754425952, 9.557293839991645149513140061386, 11.98835428651433940881288822793, 13.65506776472535092069837079094, 14.44650495679029834746295060204
