| L(s) = 1 | − 1.97·2-s + 0.955·3-s + 1.90·4-s − 3.57·5-s − 1.88·6-s − 2.56·7-s + 0.191·8-s − 2.08·9-s + 7.06·10-s − 2.18·11-s + 1.81·12-s + 1.23·13-s + 5.07·14-s − 3.41·15-s − 4.18·16-s + 2.68·17-s + 4.12·18-s − 8.30·19-s − 6.80·20-s − 2.45·21-s + 4.32·22-s + 1.65·23-s + 0.183·24-s + 7.79·25-s − 2.43·26-s − 4.86·27-s − 4.88·28-s + ⋯ |
| L(s) = 1 | − 1.39·2-s + 0.551·3-s + 0.951·4-s − 1.59·5-s − 0.770·6-s − 0.970·7-s + 0.0677·8-s − 0.695·9-s + 2.23·10-s − 0.659·11-s + 0.525·12-s + 0.341·13-s + 1.35·14-s − 0.882·15-s − 1.04·16-s + 0.650·17-s + 0.971·18-s − 1.90·19-s − 1.52·20-s − 0.535·21-s + 0.921·22-s + 0.344·23-s + 0.0373·24-s + 1.55·25-s − 0.477·26-s − 0.935·27-s − 0.923·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1717981346\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1717981346\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 \) |
| good | 2 | \( 1 + 1.97T + 2T^{2} \) |
| 3 | \( 1 - 0.955T + 3T^{2} \) |
| 5 | \( 1 + 3.57T + 5T^{2} \) |
| 7 | \( 1 + 2.56T + 7T^{2} \) |
| 11 | \( 1 + 2.18T + 11T^{2} \) |
| 13 | \( 1 - 1.23T + 13T^{2} \) |
| 17 | \( 1 - 2.68T + 17T^{2} \) |
| 19 | \( 1 + 8.30T + 19T^{2} \) |
| 23 | \( 1 - 1.65T + 23T^{2} \) |
| 29 | \( 1 + 3.71T + 29T^{2} \) |
| 31 | \( 1 + 4.21T + 31T^{2} \) |
| 37 | \( 1 + 9.61T + 37T^{2} \) |
| 41 | \( 1 + 11.1T + 41T^{2} \) |
| 47 | \( 1 - 1.54T + 47T^{2} \) |
| 53 | \( 1 - 3.99T + 53T^{2} \) |
| 59 | \( 1 - 4.71T + 59T^{2} \) |
| 61 | \( 1 - 10.3T + 61T^{2} \) |
| 67 | \( 1 + 2.17T + 67T^{2} \) |
| 71 | \( 1 - 2.35T + 71T^{2} \) |
| 73 | \( 1 - 5.37T + 73T^{2} \) |
| 79 | \( 1 + 1.60T + 79T^{2} \) |
| 83 | \( 1 - 7.02T + 83T^{2} \) |
| 89 | \( 1 - 9.42T + 89T^{2} \) |
| 97 | \( 1 + 6.48T + 97T^{2} \) |
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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
−8.850925719585071120914689744838, −8.589522682466920770256115059007, −7.954499524569660173216455237397, −7.24039536924565495836844072914, −6.51900038666396840811750913529, −5.19187236478884506702770904414, −3.89064004916787428083323146278, −3.31067901165688990841744761560, −2.11947590737127493256920083383, −0.31577694132063275575834008823,
0.31577694132063275575834008823, 2.11947590737127493256920083383, 3.31067901165688990841744761560, 3.89064004916787428083323146278, 5.19187236478884506702770904414, 6.51900038666396840811750913529, 7.24039536924565495836844072914, 7.954499524569660173216455237397, 8.589522682466920770256115059007, 8.850925719585071120914689744838
