L(s) = 1 | + (−0.0966 − 0.297i)2-s + (0.729 − 0.530i)4-s + (−0.587 − 0.809i)7-s + (−0.481 − 0.349i)8-s + (−0.453 − 0.891i)11-s + (−0.183 + 0.253i)14-s + (0.221 − 0.680i)16-s + (−0.221 + 0.221i)22-s + 1.97i·23-s + (0.809 + 0.587i)25-s + (−0.857 − 0.278i)28-s + (1.44 − 1.04i)29-s − 0.819·32-s + (−0.951 + 0.690i)37-s + 0.618i·43-s + (−0.803 − 0.409i)44-s + ⋯ |
L(s) = 1 | + (−0.0966 − 0.297i)2-s + (0.729 − 0.530i)4-s + (−0.587 − 0.809i)7-s + (−0.481 − 0.349i)8-s + (−0.453 − 0.891i)11-s + (−0.183 + 0.253i)14-s + (0.221 − 0.680i)16-s + (−0.221 + 0.221i)22-s + 1.97i·23-s + (0.809 + 0.587i)25-s + (−0.857 − 0.278i)28-s + (1.44 − 1.04i)29-s − 0.819·32-s + (−0.951 + 0.690i)37-s + 0.618i·43-s + (−0.803 − 0.409i)44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.190 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.190 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9595192121\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9595192121\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (0.587 + 0.809i)T \) |
| 11 | \( 1 + (0.453 + 0.891i)T \) |
good | 2 | \( 1 + (0.0966 + 0.297i)T + (-0.809 + 0.587i)T^{2} \) |
| 5 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 13 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 17 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 19 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 23 | \( 1 - 1.97iT - T^{2} \) |
| 29 | \( 1 + (-1.44 + 1.04i)T + (0.309 - 0.951i)T^{2} \) |
| 31 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 37 | \( 1 + (0.951 - 0.690i)T + (0.309 - 0.951i)T^{2} \) |
| 41 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 43 | \( 1 - 0.618iT - T^{2} \) |
| 47 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 53 | \( 1 + (-1.69 + 0.550i)T + (0.809 - 0.587i)T^{2} \) |
| 59 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 61 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 67 | \( 1 - 0.618T + T^{2} \) |
| 71 | \( 1 + (0.863 + 0.280i)T + (0.809 + 0.587i)T^{2} \) |
| 73 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 79 | \( 1 + (1.80 - 0.587i)T + (0.809 - 0.587i)T^{2} \) |
| 83 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (0.809 - 0.587i)T^{2} \) |
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
−10.37594545041502237276629913250, −9.951886311050152722803010337681, −8.910670978511422176849034732771, −7.74804915904844083266723294502, −6.92490468638593305273735200778, −6.09563358329533034294729899154, −5.16992503337588871658804787849, −3.65045408247538775816267399231, −2.79757146720682771443586220879, −1.16817165119937264475905190244,
2.27602869291818794927641599057, 3.01177590942118824597178671337, 4.49216267179879403719268958528, 5.64094936024933737088868088790, 6.67387597092873804679454244932, 7.13398731649512979840997806798, 8.453802646447462222938220823839, 8.792458705505487334432440055939, 10.16942601512139081162810224803, 10.68151005833140564249556600798