L(s) = 1 | + (1.34 − 0.442i)2-s − 3-s + (1.60 − 1.18i)4-s + 2.27i·5-s + (−1.34 + 0.442i)6-s − 4.97·7-s + (1.63 − 2.30i)8-s + 9-s + (1.00 + 3.05i)10-s − 2.26·11-s + (−1.60 + 1.18i)12-s + 6.26i·13-s + (−6.68 + 2.19i)14-s − 2.27i·15-s + (1.17 − 3.82i)16-s − 0.801·17-s + ⋯ |
L(s) = 1 | + (0.949 − 0.312i)2-s − 0.577·3-s + (0.804 − 0.593i)4-s + 1.01i·5-s + (−0.548 + 0.180i)6-s − 1.88·7-s + (0.578 − 0.815i)8-s + 0.333·9-s + (0.317 + 0.964i)10-s − 0.681·11-s + (−0.464 + 0.342i)12-s + 1.73i·13-s + (−1.78 + 0.587i)14-s − 0.586i·15-s + (0.294 − 0.955i)16-s − 0.194·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.310 - 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.310 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.641026 + 0.884038i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.641026 + 0.884038i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.34 + 0.442i)T \) |
| 3 | \( 1 + T \) |
| 67 | \( 1 + (2.57 - 7.77i)T \) |
good | 5 | \( 1 - 2.27iT - 5T^{2} \) |
| 7 | \( 1 + 4.97T + 7T^{2} \) |
| 11 | \( 1 + 2.26T + 11T^{2} \) |
| 13 | \( 1 - 6.26iT - 13T^{2} \) |
| 17 | \( 1 + 0.801T + 17T^{2} \) |
| 19 | \( 1 - 2.05iT - 19T^{2} \) |
| 23 | \( 1 - 7.41iT - 23T^{2} \) |
| 29 | \( 1 + 6.30T + 29T^{2} \) |
| 31 | \( 1 - 2.55T + 31T^{2} \) |
| 37 | \( 1 + 10.5T + 37T^{2} \) |
| 41 | \( 1 - 0.0427iT - 41T^{2} \) |
| 43 | \( 1 - 0.122T + 43T^{2} \) |
| 47 | \( 1 + 4.63iT - 47T^{2} \) |
| 53 | \( 1 - 6.54iT - 53T^{2} \) |
| 59 | \( 1 + 4.77iT - 59T^{2} \) |
| 61 | \( 1 + 10.5iT - 61T^{2} \) |
| 71 | \( 1 - 4.26iT - 71T^{2} \) |
| 73 | \( 1 - 12.0T + 73T^{2} \) |
| 79 | \( 1 - 1.63T + 79T^{2} \) |
| 83 | \( 1 - 5.62iT - 83T^{2} \) |
| 89 | \( 1 - 11.1T + 89T^{2} \) |
| 97 | \( 1 + 14.5iT - 97T^{2} \) |
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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
−10.63405165362110205040846048949, −9.914644093530665761745350025944, −9.276609034340949940249262066695, −7.31790319068582852576972186078, −6.79918904931740101535280618050, −6.19659976688907755340744224819, −5.29322838895890345437915954899, −3.88359140159627655290549946572, −3.27440453299787642947007439797, −2.05562233641326792885421349448,
0.39159835404168123447899487524, 2.66802054512449387906177496605, 3.58619488609435977279060081976, 4.82244448134597527563410838478, 5.52705874031497066416610197556, 6.28601641336371651747015522691, 7.12817774139519117846969351256, 8.169532946363812164389808065682, 9.115469304096980608453179000111, 10.29817090735073282354567325811