| L(s) = 1 | + (1.91 − 1.41i)2-s + (−0.435 + 2.30i)3-s + (1.08 − 3.51i)4-s + (−0.384 + 2.20i)5-s + (2.42 + 5.02i)6-s + (1.41 + 2.23i)7-s + (−1.32 − 3.77i)8-s + (−2.31 − 0.908i)9-s + (2.38 + 4.76i)10-s + (−1.66 − 4.23i)11-s + (7.62 + 4.02i)12-s + (−0.0800 − 0.710i)13-s + (5.87 + 2.27i)14-s + (−4.90 − 1.84i)15-s + (−1.80 − 1.22i)16-s + (−0.159 − 4.26i)17-s + ⋯ |
| L(s) = 1 | + (1.35 − 1.00i)2-s + (−0.251 + 1.32i)3-s + (0.542 − 1.75i)4-s + (−0.171 + 0.985i)5-s + (0.988 + 2.05i)6-s + (0.535 + 0.844i)7-s + (−0.467 − 1.33i)8-s + (−0.771 − 0.302i)9-s + (0.752 + 1.50i)10-s + (−0.501 − 1.27i)11-s + (2.20 + 1.16i)12-s + (−0.0221 − 0.196i)13-s + (1.57 + 0.608i)14-s + (−1.26 − 0.476i)15-s + (−0.450 − 0.307i)16-s + (−0.0387 − 1.03i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.25966 + 0.143102i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.25966 + 0.143102i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.384 - 2.20i)T \) |
| 7 | \( 1 + (-1.41 - 2.23i)T \) |
| good | 2 | \( 1 + (-1.91 + 1.41i)T + (0.589 - 1.91i)T^{2} \) |
| 3 | \( 1 + (0.435 - 2.30i)T + (-2.79 - 1.09i)T^{2} \) |
| 11 | \( 1 + (1.66 + 4.23i)T + (-8.06 + 7.48i)T^{2} \) |
| 13 | \( 1 + (0.0800 + 0.710i)T + (-12.6 + 2.89i)T^{2} \) |
| 17 | \( 1 + (0.159 + 4.26i)T + (-16.9 + 1.27i)T^{2} \) |
| 19 | \( 1 + (3.03 + 5.26i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-7.32 - 0.274i)T + (22.9 + 1.71i)T^{2} \) |
| 29 | \( 1 + (-2.15 - 0.491i)T + (26.1 + 12.5i)T^{2} \) |
| 31 | \( 1 + (0.305 + 0.176i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3.92 - 7.42i)T + (-20.8 - 30.5i)T^{2} \) |
| 41 | \( 1 + (2.60 - 5.40i)T + (-25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (3.95 + 1.38i)T + (33.6 + 26.8i)T^{2} \) |
| 47 | \( 1 + (1.54 + 2.09i)T + (-13.8 + 44.9i)T^{2} \) |
| 53 | \( 1 + (1.40 + 2.65i)T + (-29.8 + 43.7i)T^{2} \) |
| 59 | \( 1 + (-0.936 + 12.4i)T + (-58.3 - 8.79i)T^{2} \) |
| 61 | \( 1 + (-0.327 - 1.06i)T + (-50.4 + 34.3i)T^{2} \) |
| 67 | \( 1 + (1.89 + 7.08i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (-2.59 - 11.3i)T + (-63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (-0.733 + 0.993i)T + (-21.5 - 69.7i)T^{2} \) |
| 79 | \( 1 + (7.95 - 4.59i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.50 - 0.394i)T + (80.9 + 18.4i)T^{2} \) |
| 89 | \( 1 + (4.59 - 11.6i)T + (-65.2 - 60.5i)T^{2} \) |
| 97 | \( 1 + (-7.13 - 7.13i)T + 97iT^{2} \) |
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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
−11.71341877534358904435880288509, −11.16355667654261469874920361633, −10.77570899973409889908362989041, −9.720052490513151253489759097165, −8.480496682538987557191980639130, −6.57920968902557183274601984015, −5.28583544166452602195237251142, −4.81569690693363214304672073560, −3.35373520444035317427249239504, −2.72016391958526723763989471887,
1.68149872532433038589959924370, 4.02431153359905174402146420203, 4.88740586576271838127144659567, 5.97165076397140377400757106824, 7.09030443963275320269309301952, 7.59059724334902326869084794581, 8.503282181152499404816959062714, 10.36462407995500392574483841359, 11.82694268334045238079669771801, 12.67694124321953178567735389394
