| L(s) = 1 | + (−1.21 − 1.77i)2-s + (1.72 + 1.85i)3-s + (−0.963 + 2.45i)4-s + (−1.96 + 1.07i)5-s + (1.21 − 5.32i)6-s + (−1.53 + 2.15i)7-s + (1.34 − 0.306i)8-s + (−0.255 + 3.41i)9-s + (4.28 + 2.18i)10-s + (−0.129 − 1.72i)11-s + (−6.22 + 2.44i)12-s + (−0.896 + 1.86i)13-s + (5.69 + 0.106i)14-s + (−5.37 − 1.79i)15-s + (1.69 + 1.57i)16-s + (−0.558 + 3.70i)17-s + ⋯ |
| L(s) = 1 | + (−0.857 − 1.25i)2-s + (0.995 + 1.07i)3-s + (−0.481 + 1.22i)4-s + (−0.877 + 0.479i)5-s + (0.495 − 2.17i)6-s + (−0.578 + 0.815i)7-s + (0.474 − 0.108i)8-s + (−0.0852 + 1.13i)9-s + (1.35 + 0.692i)10-s + (−0.0390 − 0.521i)11-s + (−1.79 + 0.705i)12-s + (−0.248 + 0.516i)13-s + (1.52 + 0.0285i)14-s + (−1.38 − 0.463i)15-s + (0.424 + 0.393i)16-s + (−0.135 + 0.898i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.430 - 0.902i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.430 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.593900 + 0.374583i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.593900 + 0.374583i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (1.96 - 1.07i)T \) |
| 7 | \( 1 + (1.53 - 2.15i)T \) |
| good | 2 | \( 1 + (1.21 + 1.77i)T + (-0.730 + 1.86i)T^{2} \) |
| 3 | \( 1 + (-1.72 - 1.85i)T + (-0.224 + 2.99i)T^{2} \) |
| 11 | \( 1 + (0.129 + 1.72i)T + (-10.8 + 1.63i)T^{2} \) |
| 13 | \( 1 + (0.896 - 1.86i)T + (-8.10 - 10.1i)T^{2} \) |
| 17 | \( 1 + (0.558 - 3.70i)T + (-16.2 - 5.01i)T^{2} \) |
| 19 | \( 1 + (3.06 - 5.31i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.162 - 1.07i)T + (-21.9 + 6.77i)T^{2} \) |
| 29 | \( 1 + (-5.49 + 6.88i)T + (-6.45 - 28.2i)T^{2} \) |
| 31 | \( 1 + (-1.40 - 2.43i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.17 - 0.461i)T + (27.1 - 25.1i)T^{2} \) |
| 41 | \( 1 + (-1.48 - 6.49i)T + (-36.9 + 17.7i)T^{2} \) |
| 43 | \( 1 + (-11.5 - 2.63i)T + (38.7 + 18.6i)T^{2} \) |
| 47 | \( 1 + (5.24 + 7.69i)T + (-17.1 + 43.7i)T^{2} \) |
| 53 | \( 1 + (11.9 + 4.69i)T + (38.8 + 36.0i)T^{2} \) |
| 59 | \( 1 + (-9.06 + 2.79i)T + (48.7 - 33.2i)T^{2} \) |
| 61 | \( 1 + (-1.68 - 4.29i)T + (-44.7 + 41.4i)T^{2} \) |
| 67 | \( 1 + (3.69 - 2.13i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-3.56 - 4.46i)T + (-15.7 + 69.2i)T^{2} \) |
| 73 | \( 1 + (2.14 - 3.14i)T + (-26.6 - 67.9i)T^{2} \) |
| 79 | \( 1 + (2.00 - 3.46i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-5.87 - 12.1i)T + (-51.7 + 64.8i)T^{2} \) |
| 89 | \( 1 + (0.147 - 1.96i)T + (-88.0 - 13.2i)T^{2} \) |
| 97 | \( 1 + 11.3iT - 97T^{2} \) |
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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
−11.96172639353256850254916208541, −11.09352163393240599116022599682, −10.16264270062553678423744215279, −9.624860769685177110812158494974, −8.481076712417611076826870965279, −8.232748922322315867418440044758, −6.25020800067751025860147166605, −4.17079458293568795860354035741, −3.35951752801739157595693163741, −2.41578558805624804082353140388,
0.65831721359374480058829655921, 2.98679957025720629548592036106, 4.73447922250053100162038522025, 6.56199289395199071781474409715, 7.29209847163689098708090128307, 7.72547908371465193837939944408, 8.738980433857630445186046744414, 9.352082800700282074616807941756, 10.72292572911357241071835612900, 12.32677104951090292082631374801
