| L(s) = 1 | − 2.72·2-s + 5.42·4-s − 5-s − 4.53·7-s − 9.32·8-s + 2.72·10-s − 4.77·11-s + 13-s + 12.3·14-s + 14.5·16-s − 4.35·17-s − 7.13·19-s − 5.42·20-s + 13.0·22-s − 6.97·23-s + 25-s − 2.72·26-s − 24.6·28-s + 0.650·29-s − 2.33·31-s − 21.0·32-s + 11.8·34-s + 4.53·35-s + 2.41·37-s + 19.4·38-s + 9.32·40-s + 6.66·41-s + ⋯ |
| L(s) = 1 | − 1.92·2-s + 2.71·4-s − 0.447·5-s − 1.71·7-s − 3.29·8-s + 0.861·10-s − 1.43·11-s + 0.277·13-s + 3.30·14-s + 3.63·16-s − 1.05·17-s − 1.63·19-s − 1.21·20-s + 2.77·22-s − 1.45·23-s + 0.200·25-s − 0.534·26-s − 4.64·28-s + 0.120·29-s − 0.419·31-s − 3.71·32-s + 2.03·34-s + 0.766·35-s + 0.396·37-s + 3.15·38-s + 1.47·40-s + 1.04·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.08872066281\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08872066281\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 + 2.72T + 2T^{2} \) |
| 7 | \( 1 + 4.53T + 7T^{2} \) |
| 11 | \( 1 + 4.77T + 11T^{2} \) |
| 17 | \( 1 + 4.35T + 17T^{2} \) |
| 19 | \( 1 + 7.13T + 19T^{2} \) |
| 23 | \( 1 + 6.97T + 23T^{2} \) |
| 29 | \( 1 - 0.650T + 29T^{2} \) |
| 31 | \( 1 + 2.33T + 31T^{2} \) |
| 37 | \( 1 - 2.41T + 37T^{2} \) |
| 41 | \( 1 - 6.66T + 41T^{2} \) |
| 43 | \( 1 + 10.5T + 43T^{2} \) |
| 47 | \( 1 - 4.79T + 47T^{2} \) |
| 53 | \( 1 + 2.54T + 53T^{2} \) |
| 59 | \( 1 - 2.05T + 59T^{2} \) |
| 61 | \( 1 - 8.41T + 61T^{2} \) |
| 67 | \( 1 - 4.76T + 67T^{2} \) |
| 71 | \( 1 + 3.01T + 71T^{2} \) |
| 73 | \( 1 + 4.07T + 73T^{2} \) |
| 79 | \( 1 + 13.1T + 79T^{2} \) |
| 83 | \( 1 - 0.290T + 83T^{2} \) |
| 89 | \( 1 - 11.1T + 89T^{2} \) |
| 97 | \( 1 + 8.90T + 97T^{2} \) |
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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
−9.297691214029535678749617031253, −8.507728992834570057773716319119, −8.016209389408710417630157144168, −7.05745685875715171822132556377, −6.48256201139672726162684835241, −5.79068275173769611721830300560, −4.00582065128505387205262309446, −2.84307491662419737590230936311, −2.15450483439928712425993693785, −0.24665492472395169415004765773,
0.24665492472395169415004765773, 2.15450483439928712425993693785, 2.84307491662419737590230936311, 4.00582065128505387205262309446, 5.79068275173769611721830300560, 6.48256201139672726162684835241, 7.05745685875715171822132556377, 8.016209389408710417630157144168, 8.507728992834570057773716319119, 9.297691214029535678749617031253
