| L(s) = 1 | + (−1.32 + 0.495i)2-s + (1.60 + 0.662i)3-s + (1.50 − 1.31i)4-s + (0.462 + 2.18i)5-s + (−2.44 − 0.0845i)6-s + 2.28·7-s + (−1.34 + 2.48i)8-s + (2.12 + 2.11i)9-s + (−1.69 − 2.66i)10-s + (−2.72 − 1.98i)11-s + (3.28 − 1.10i)12-s + (0.498 + 0.686i)13-s + (−3.02 + 1.13i)14-s + (−0.708 + 3.80i)15-s + (0.556 − 3.96i)16-s + (−0.267 + 0.823i)17-s + ⋯ |
| L(s) = 1 | + (−0.936 + 0.350i)2-s + (0.924 + 0.382i)3-s + (0.754 − 0.656i)4-s + (0.206 + 0.978i)5-s + (−0.999 − 0.0345i)6-s + 0.864·7-s + (−0.477 + 0.878i)8-s + (0.707 + 0.706i)9-s + (−0.536 − 0.844i)10-s + (−0.822 − 0.597i)11-s + (0.948 − 0.317i)12-s + (0.138 + 0.190i)13-s + (−0.809 + 0.302i)14-s + (−0.183 + 0.983i)15-s + (0.139 − 0.990i)16-s + (−0.0648 + 0.199i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.322 - 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.322 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.02636 + 0.734638i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.02636 + 0.734638i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.32 - 0.495i)T \) |
| 3 | \( 1 + (-1.60 - 0.662i)T \) |
| 5 | \( 1 + (-0.462 - 2.18i)T \) |
| good | 7 | \( 1 - 2.28T + 7T^{2} \) |
| 11 | \( 1 + (2.72 + 1.98i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (-0.498 - 0.686i)T + (-4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (0.267 - 0.823i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-0.177 - 0.0578i)T + (15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-3.95 + 5.44i)T + (-7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (3.23 - 1.05i)T + (23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (6.19 + 2.01i)T + (25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-5.08 - 6.99i)T + (-11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-5.24 - 7.21i)T + (-12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 2.72T + 43T^{2} \) |
| 47 | \( 1 + (-9.30 + 3.02i)T + (38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (3.90 + 12.0i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-3.25 + 2.36i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (9.18 + 6.67i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (-0.869 + 2.67i)T + (-54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (4.05 + 12.4i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-2.89 + 3.99i)T + (-22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (-5.38 + 1.75i)T + (63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (11.0 + 3.59i)T + (67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (0.0803 - 0.110i)T + (-27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-14.4 + 4.70i)T + (78.4 - 57.0i)T^{2} \) |
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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
−11.28279188441766910637962915711, −10.84741535509931601019725856439, −9.996628641895335594660432268401, −9.010316468361043897568040733197, −8.103539891691699832214239203590, −7.46832194169524981722201688228, −6.28350363424245819343557084945, −4.93808169003877307691929847569, −3.14203821219549207340705395426, −2.01459562763828154097326599006,
1.33424842775584592055015847930, 2.46891174461267888748731739268, 4.08214794985979084490951896085, 5.56391944012357171542103674566, 7.41822887561318229609658485605, 7.72656047248998855938151335389, 8.906843713485533857805908526230, 9.304381429672514066799304637042, 10.46759257033476467217347737356, 11.50303215699454581976415155333
