L(s) = 1 | − 28.5·3-s + (−5.81 − 55.5i)5-s − 92.3i·7-s + 571.·9-s − 541. i·11-s + 782.·13-s + (165. + 1.58e3i)15-s + 1.55e3i·17-s − 484. i·19-s + 2.63e3i·21-s + 1.09e3i·23-s + (−3.05e3 + 646. i)25-s − 9.37e3·27-s + 597. i·29-s + 8.86e3·31-s + ⋯ |
L(s) = 1 | − 1.83·3-s + (−0.104 − 0.994i)5-s − 0.712i·7-s + 2.35·9-s − 1.34i·11-s + 1.28·13-s + (0.190 + 1.82i)15-s + 1.30i·17-s − 0.307i·19-s + 1.30i·21-s + 0.432i·23-s + (−0.978 + 0.206i)25-s − 2.47·27-s + 0.131i·29-s + 1.65·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.357 + 0.933i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.357 + 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.178255591\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.178255591\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (5.81 + 55.5i)T \) |
good | 3 | \( 1 + 28.5T + 243T^{2} \) |
| 7 | \( 1 + 92.3iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 541. iT - 1.61e5T^{2} \) |
| 13 | \( 1 - 782.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.55e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 484. iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 1.09e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 597. iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 8.86e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.36e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.45e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.82e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.79e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 1.65e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.44e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 - 2.45e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 - 2.68e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.80e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.06e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 2.81e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 2.45e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.17e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.77e4iT - 8.58e9T^{2} \) |
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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
−11.00192003393488386309829141220, −9.984611988610378845401937282661, −8.695221882264239209399399769324, −7.70375600790689082836613161595, −6.09354581733175853529093065994, −6.02536767910065945974712826763, −4.64923879097424342405154247116, −3.84833922434319025411829689884, −1.12960743037922320744119293106, −0.74190205412005657334105676040,
0.789386651532782260796325753724, 2.37019538467195793629830930018, 4.13045530191499530450879607279, 5.16439649180927758348644983750, 6.21009862337512040257131561419, 6.73726598383257128227104366903, 7.80208869517816908256767654650, 9.518139471736587956965369136771, 10.25538954063385179380770466852, 11.15847066230491538442325895299