L(s) = 1 | + (1.52 − 1.63i)5-s − 2.82i·7-s − 1.05·11-s − i·13-s − 7.14i·17-s + 6.61·19-s − 3.49i·23-s + (−0.332 − 4.98i)25-s + 4.82·29-s − 9.65·31-s + (−4.61 − 4.32i)35-s + 8.11i·37-s + 3.26·41-s + 7.44i·43-s − 11.3i·47-s + ⋯ |
L(s) = 1 | + (0.683 − 0.730i)5-s − 1.06i·7-s − 0.318·11-s − 0.277i·13-s − 1.73i·17-s + 1.51·19-s − 0.728i·23-s + (−0.0664 − 0.997i)25-s + 0.896·29-s − 1.73·31-s + (−0.780 − 0.730i)35-s + 1.33i·37-s + 0.510·41-s + 1.13i·43-s − 1.65i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.683 + 0.730i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.683 + 0.730i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.928366458\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.928366458\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.52 + 1.63i)T \) |
| 13 | \( 1 + iT \) |
good | 7 | \( 1 + 2.82iT - 7T^{2} \) |
| 11 | \( 1 + 1.05T + 11T^{2} \) |
| 17 | \( 1 + 7.14iT - 17T^{2} \) |
| 19 | \( 1 - 6.61T + 19T^{2} \) |
| 23 | \( 1 + 3.49iT - 23T^{2} \) |
| 29 | \( 1 - 4.82T + 29T^{2} \) |
| 31 | \( 1 + 9.65T + 31T^{2} \) |
| 37 | \( 1 - 8.11iT - 37T^{2} \) |
| 41 | \( 1 - 3.26T + 41T^{2} \) |
| 43 | \( 1 - 7.44iT - 43T^{2} \) |
| 47 | \( 1 + 11.3iT - 47T^{2} \) |
| 53 | \( 1 - 4.53iT - 53T^{2} \) |
| 59 | \( 1 - 1.05T + 59T^{2} \) |
| 61 | \( 1 + 7.97T + 61T^{2} \) |
| 67 | \( 1 - 1.46iT - 67T^{2} \) |
| 71 | \( 1 + 0.845T + 71T^{2} \) |
| 73 | \( 1 + 9.28iT - 73T^{2} \) |
| 79 | \( 1 + 6.85T + 79T^{2} \) |
| 83 | \( 1 - 7.97iT - 83T^{2} \) |
| 89 | \( 1 - 0.139T + 89T^{2} \) |
| 97 | \( 1 + 6.93iT - 97T^{2} \) |
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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
−7.919594688461571171934702095075, −7.35844061559144918894569242519, −6.69126291915292812367920851214, −5.70663668014923120428059313885, −5.01251807000594883752029422140, −4.52923251221178487018877220867, −3.37633073819091740884116623124, −2.59138011666844579531788559869, −1.31331424913017253034852465512, −0.53720958167864161685784998010,
1.51919696080820127812169142954, 2.26055987208327737256488728166, 3.14623011008978271722492022281, 3.89520460169515740766234321917, 5.17500909928742344101642611359, 5.75699315667655305166563039712, 6.15985132841650751254272630290, 7.21525052107068006473882320706, 7.70689100547020421233568229406, 8.746758902717631829832966882956