| L(s) = 1 | + (1.52 + 1.04i)2-s + (−1.14 − 1.06i)3-s + (0.514 + 1.31i)4-s + (−1.32 + 0.410i)5-s + (−0.644 − 2.82i)6-s + (−1.73 + 1.99i)7-s + (0.242 − 1.06i)8-s + (−0.0403 − 0.538i)9-s + (−2.45 − 0.757i)10-s + (−0.215 + 2.87i)11-s + (0.807 − 2.05i)12-s + (4.03 − 1.94i)13-s + (−4.72 + 1.23i)14-s + (1.96 + 0.947i)15-s + (3.54 − 3.28i)16-s + (−0.476 + 0.0718i)17-s + ⋯ |
| L(s) = 1 | + (1.07 + 0.735i)2-s + (−0.663 − 0.615i)3-s + (0.257 + 0.655i)4-s + (−0.594 + 0.183i)5-s + (−0.263 − 1.15i)6-s + (−0.656 + 0.754i)7-s + (0.0859 − 0.376i)8-s + (−0.0134 − 0.179i)9-s + (−0.776 − 0.239i)10-s + (−0.0649 + 0.866i)11-s + (0.232 − 0.593i)12-s + (1.12 − 0.539i)13-s + (−1.26 + 0.330i)14-s + (0.507 + 0.244i)15-s + (0.885 − 0.821i)16-s + (−0.115 + 0.0174i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.901 - 0.432i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.901 - 0.432i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.996250 + 0.226778i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.996250 + 0.226778i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (1.73 - 1.99i)T \) |
| good | 2 | \( 1 + (-1.52 - 1.04i)T + (0.730 + 1.86i)T^{2} \) |
| 3 | \( 1 + (1.14 + 1.06i)T + (0.224 + 2.99i)T^{2} \) |
| 5 | \( 1 + (1.32 - 0.410i)T + (4.13 - 2.81i)T^{2} \) |
| 11 | \( 1 + (0.215 - 2.87i)T + (-10.8 - 1.63i)T^{2} \) |
| 13 | \( 1 + (-4.03 + 1.94i)T + (8.10 - 10.1i)T^{2} \) |
| 17 | \( 1 + (0.476 - 0.0718i)T + (16.2 - 5.01i)T^{2} \) |
| 19 | \( 1 + (-1.90 - 3.30i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (6.73 + 1.01i)T + (21.9 + 6.77i)T^{2} \) |
| 29 | \( 1 + (-1.45 - 1.81i)T + (-6.45 + 28.2i)T^{2} \) |
| 31 | \( 1 + (-3.94 + 6.82i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.50 + 8.92i)T + (-27.1 - 25.1i)T^{2} \) |
| 41 | \( 1 + (1.56 - 6.84i)T + (-36.9 - 17.7i)T^{2} \) |
| 43 | \( 1 + (-0.546 - 2.39i)T + (-38.7 + 18.6i)T^{2} \) |
| 47 | \( 1 + (9.25 + 6.30i)T + (17.1 + 43.7i)T^{2} \) |
| 53 | \( 1 + (-1.96 - 4.99i)T + (-38.8 + 36.0i)T^{2} \) |
| 59 | \( 1 + (-3.57 - 1.10i)T + (48.7 + 33.2i)T^{2} \) |
| 61 | \( 1 + (-0.0147 + 0.0376i)T + (-44.7 - 41.4i)T^{2} \) |
| 67 | \( 1 + (0.534 - 0.926i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-4.21 + 5.28i)T + (-15.7 - 69.2i)T^{2} \) |
| 73 | \( 1 + (6.78 - 4.62i)T + (26.6 - 67.9i)T^{2} \) |
| 79 | \( 1 + (-6.91 - 11.9i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (0.465 + 0.224i)T + (51.7 + 64.8i)T^{2} \) |
| 89 | \( 1 + (-0.0586 - 0.782i)T + (-88.0 + 13.2i)T^{2} \) |
| 97 | \( 1 - 9.61T + 97T^{2} \) |
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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
−15.58027493717256227989950456240, −14.77532940103341779064275747255, −13.31423587618541372409414531761, −12.49024282634597482635413783747, −11.66417025467481460760123044114, −9.815327149449058089915040947132, −7.81013656638924003115736419232, −6.46076070225792443816457011601, −5.70824761723932592336545662576, −3.80031813593763490651188946305,
3.54443443509831531332488150940, 4.60808636581104831983482198250, 6.14536829545925118684541609247, 8.242922484392453567372611839659, 10.19802094993201140841594469690, 11.20804464995068844957661144091, 11.89372937657449874418180971383, 13.42279059802525824272528162359, 13.89777956941890621305440476385, 15.81859935134223206673588652362
