| L(s) = 1 | + (0.406 + 0.913i)2-s + (−0.383 − 0.345i)3-s + (−0.669 + 0.743i)4-s + (2.15 − 0.588i)5-s + (0.159 − 0.490i)6-s + (0.792 − 2.52i)7-s + (−0.951 − 0.309i)8-s + (−0.285 − 2.71i)9-s + (1.41 + 1.73i)10-s + (0.0983 − 0.935i)11-s + (0.513 − 0.0539i)12-s + (3.32 + 4.57i)13-s + (2.62 − 0.303i)14-s + (−1.03 − 0.519i)15-s + (−0.104 − 0.994i)16-s + (−1.09 − 5.16i)17-s + ⋯ |
| L(s) = 1 | + (0.287 + 0.645i)2-s + (−0.221 − 0.199i)3-s + (−0.334 + 0.371i)4-s + (0.964 − 0.262i)5-s + (0.0650 − 0.200i)6-s + (0.299 − 0.954i)7-s + (−0.336 − 0.109i)8-s + (−0.0952 − 0.906i)9-s + (0.447 + 0.547i)10-s + (0.0296 − 0.281i)11-s + (0.148 − 0.0155i)12-s + (0.922 + 1.27i)13-s + (0.702 − 0.0809i)14-s + (−0.266 − 0.134i)15-s + (−0.0261 − 0.248i)16-s + (−0.266 − 1.25i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0610i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0610i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.65623 + 0.0506267i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.65623 + 0.0506267i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.406 - 0.913i)T \) |
| 5 | \( 1 + (-2.15 + 0.588i)T \) |
| 7 | \( 1 + (-0.792 + 2.52i)T \) |
| good | 3 | \( 1 + (0.383 + 0.345i)T + (0.313 + 2.98i)T^{2} \) |
| 11 | \( 1 + (-0.0983 + 0.935i)T + (-10.7 - 2.28i)T^{2} \) |
| 13 | \( 1 + (-3.32 - 4.57i)T + (-4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (1.09 + 5.16i)T + (-15.5 + 6.91i)T^{2} \) |
| 19 | \( 1 + (-0.322 - 0.358i)T + (-1.98 + 18.8i)T^{2} \) |
| 23 | \( 1 + (-1.83 - 4.11i)T + (-15.3 + 17.0i)T^{2} \) |
| 29 | \( 1 + (-2.72 - 8.37i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (7.73 - 1.64i)T + (28.3 - 12.6i)T^{2} \) |
| 37 | \( 1 + (-1.81 + 0.191i)T + (36.1 - 7.69i)T^{2} \) |
| 41 | \( 1 + (-3.66 + 2.66i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + 1.60iT - 43T^{2} \) |
| 47 | \( 1 + (-0.268 + 1.26i)T + (-42.9 - 19.1i)T^{2} \) |
| 53 | \( 1 + (-5.57 - 5.01i)T + (5.54 + 52.7i)T^{2} \) |
| 59 | \( 1 + (13.2 + 5.89i)T + (39.4 + 43.8i)T^{2} \) |
| 61 | \( 1 + (12.7 - 5.68i)T + (40.8 - 45.3i)T^{2} \) |
| 67 | \( 1 + (-1.92 - 9.06i)T + (-61.2 + 27.2i)T^{2} \) |
| 71 | \( 1 + (3.65 + 11.2i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-0.646 - 0.0679i)T + (71.4 + 15.1i)T^{2} \) |
| 79 | \( 1 + (6.80 + 1.44i)T + (72.1 + 32.1i)T^{2} \) |
| 83 | \( 1 + (-4.89 - 1.58i)T + (67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (4.03 - 1.79i)T + (59.5 - 66.1i)T^{2} \) |
| 97 | \( 1 + (-1.17 + 0.382i)T + (78.4 - 57.0i)T^{2} \) |
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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
−11.53146447615242832751392921872, −10.65901844738067763011847232601, −9.206912201737350502198712066354, −9.007301312817078561499649928094, −7.33994172318260238099888452133, −6.69880346025596260451629157318, −5.75606859614151700220351578818, −4.66776054885478776212597501032, −3.43897189679979288639208651152, −1.31332048667573558558514398343,
1.84642746367443469290969034521, 2.88594946543252424687402683134, 4.52219187409555496022037723885, 5.65981444987181471837086812663, 6.11470780026801142946687445734, 7.951972689055303206989419607505, 8.824234536317590619831383943700, 9.903714239809274226627418617701, 10.72279986435017100337811748518, 11.19128754965999231123074830365
