L(s) = 1 | − 1.77·2-s + (1.65 + 0.525i)3-s + 1.15·4-s + 3.85i·5-s + (−2.93 − 0.933i)6-s + (1.56 − 2.12i)7-s + 1.50·8-s + (2.44 + 1.73i)9-s − 6.84i·10-s − 2.15·11-s + (1.90 + 0.606i)12-s + (−2.48 + 2.61i)13-s + (−2.78 + 3.78i)14-s + (−2.02 + 6.36i)15-s − 4.97·16-s − 1.53·17-s + ⋯ |
L(s) = 1 | − 1.25·2-s + (0.952 + 0.303i)3-s + 0.576·4-s + 1.72i·5-s + (−1.19 − 0.381i)6-s + (0.593 − 0.804i)7-s + 0.531·8-s + (0.815 + 0.578i)9-s − 2.16i·10-s − 0.649·11-s + (0.549 + 0.175i)12-s + (−0.688 + 0.725i)13-s + (−0.745 + 1.01i)14-s + (−0.523 + 1.64i)15-s − 1.24·16-s − 0.371·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.131 - 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.131 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.678515 + 0.594638i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.678515 + 0.594638i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.65 - 0.525i)T \) |
| 7 | \( 1 + (-1.56 + 2.12i)T \) |
| 13 | \( 1 + (2.48 - 2.61i)T \) |
good | 2 | \( 1 + 1.77T + 2T^{2} \) |
| 5 | \( 1 - 3.85iT - 5T^{2} \) |
| 11 | \( 1 + 2.15T + 11T^{2} \) |
| 17 | \( 1 + 1.53T + 17T^{2} \) |
| 19 | \( 1 - 6.24T + 19T^{2} \) |
| 23 | \( 1 - 0.929iT - 23T^{2} \) |
| 29 | \( 1 - 2.00iT - 29T^{2} \) |
| 31 | \( 1 - 0.380T + 31T^{2} \) |
| 37 | \( 1 - 8.77iT - 37T^{2} \) |
| 41 | \( 1 + 2.58iT - 41T^{2} \) |
| 43 | \( 1 - 5.97T + 43T^{2} \) |
| 47 | \( 1 + 1.59iT - 47T^{2} \) |
| 53 | \( 1 + 12.4iT - 53T^{2} \) |
| 59 | \( 1 - 0.317iT - 59T^{2} \) |
| 61 | \( 1 + 1.21iT - 61T^{2} \) |
| 67 | \( 1 + 11.0iT - 67T^{2} \) |
| 71 | \( 1 + 7.83T + 71T^{2} \) |
| 73 | \( 1 + 1.37T + 73T^{2} \) |
| 79 | \( 1 - 10.4T + 79T^{2} \) |
| 83 | \( 1 + 7.59iT - 83T^{2} \) |
| 89 | \( 1 - 4.75iT - 89T^{2} \) |
| 97 | \( 1 - 13.5T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.55079262430021101537918183022, −10.73986420019936825792039901729, −10.10824696222412465201104799778, −9.463312661284577955846887848359, −8.175908020936996883405265125851, −7.41146865882512327437384708247, −6.95314068916535005626114010927, −4.71743451219271011031313990858, −3.30828940981271904074618320024, −2.01879505182641099005345962843,
1.01938754037419571432028315885, 2.37804995497929602958442361814, 4.50100749402729524664434238835, 5.48206696650387404349878862511, 7.64972948584783602877908904261, 7.911676791902143007863035260627, 9.013921938906172086754891343988, 9.207713188838001273137504700917, 10.30222261668978089576768932606, 11.77364627973191367535555941656