| L(s) = 1 | + (−1.41 + 1.41i)2-s − 4.00i·4-s + (−0.123 − 0.123i)7-s + (5.65 + 5.65i)8-s − 4.24i·11-s + (5.62 − 5.62i)13-s + 0.349·14-s − 16.0·16-s + (−34.4 + 34.4i)17-s − 110. i·19-s + (6 + 6i)22-s + (111. + 111. i)23-s + 15.9i·26-s + (−0.494 + 0.494i)28-s − 53.0·29-s + ⋯ |
| L(s) = 1 | + (−0.499 + 0.499i)2-s − 0.500i·4-s + (−0.00668 − 0.00668i)7-s + (0.250 + 0.250i)8-s − 0.116i·11-s + (0.120 − 0.120i)13-s + 0.00668·14-s − 0.250·16-s + (−0.491 + 0.491i)17-s − 1.33i·19-s + (0.0581 + 0.0581i)22-s + (1.01 + 1.01i)23-s + 0.120i·26-s + (−0.00334 + 0.00334i)28-s − 0.339·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.279 + 0.960i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.279 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8888822339\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8888822339\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.41 - 1.41i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + (0.123 + 0.123i)T + 343iT^{2} \) |
| 11 | \( 1 + 4.24iT - 1.33e3T^{2} \) |
| 13 | \( 1 + (-5.62 + 5.62i)T - 2.19e3iT^{2} \) |
| 17 | \( 1 + (34.4 - 34.4i)T - 4.91e3iT^{2} \) |
| 19 | \( 1 + 110. iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (-111. - 111. i)T + 1.21e4iT^{2} \) |
| 29 | \( 1 + 53.0T + 2.43e4T^{2} \) |
| 31 | \( 1 + 201.T + 2.97e4T^{2} \) |
| 37 | \( 1 + (169. + 169. i)T + 5.06e4iT^{2} \) |
| 41 | \( 1 + 371. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + (-202. + 202. i)T - 7.95e4iT^{2} \) |
| 47 | \( 1 + (106. - 106. i)T - 1.03e5iT^{2} \) |
| 53 | \( 1 + (-243. - 243. i)T + 1.48e5iT^{2} \) |
| 59 | \( 1 - 57.2T + 2.05e5T^{2} \) |
| 61 | \( 1 + 609.T + 2.26e5T^{2} \) |
| 67 | \( 1 + (340. + 340. i)T + 3.00e5iT^{2} \) |
| 71 | \( 1 + 990. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-847. + 847. i)T - 3.89e5iT^{2} \) |
| 79 | \( 1 + 436. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + (127. + 127. i)T + 5.71e5iT^{2} \) |
| 89 | \( 1 - 1.23e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-202. - 202. i)T + 9.12e5iT^{2} \) |
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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
−10.61684413113353605451376282602, −9.166103437003762274774801090300, −8.961179313684343303756043067980, −7.60364070386354183945222259604, −6.96910774692143890126759406102, −5.83400445801737163257820491627, −4.90753552591045827030889519324, −3.51210680343995733053595179153, −1.94479859461969173932825639779, −0.36069029635458941821008406895,
1.26641665457677475905099528552, 2.60174073292973557427473266935, 3.81614979999797044370577567753, 4.98752521185767329141493923099, 6.31512767070032848127580905037, 7.31700804674597330397099704826, 8.292201564144435184789533406697, 9.127790147608863519777208972117, 9.979419451265410173302705219083, 10.84527258956152398480912311185
