L(s) = 1 | + i·2-s + i·3-s − 4-s + (−2 + i)5-s − 6-s − 4i·7-s − i·8-s − 9-s + (−1 − 2i)10-s − 2·11-s − i·12-s − 4i·13-s + 4·14-s + (−1 − 2i)15-s + 16-s − i·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + 0.577i·3-s − 0.5·4-s + (−0.894 + 0.447i)5-s − 0.408·6-s − 1.51i·7-s − 0.353i·8-s − 0.333·9-s + (−0.316 − 0.632i)10-s − 0.603·11-s − 0.288i·12-s − 1.10i·13-s + 1.06·14-s + (−0.258 − 0.516i)15-s + 0.250·16-s − 0.242i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 510 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 510 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.440069 - 0.271977i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.440069 - 0.271977i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (2 - i)T \) |
| 17 | \( 1 + iT \) |
good | 7 | \( 1 + 4iT - 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 + 4T + 41T^{2} \) |
| 43 | \( 1 - 10iT - 43T^{2} \) |
| 47 | \( 1 + 12iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 + 14iT - 67T^{2} \) |
| 71 | \( 1 + 14T + 71T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−10.49974149921133142696222724514, −10.16838541135129208922861853916, −8.744815971235789548768164916395, −7.80876309504396342464846235814, −7.33116100369922786301500933470, −6.22830763168263249805995226801, −4.91689965990634807482461056148, −4.08517655392268342187749956098, −3.15735182014414183338662447762, −0.30272046569716413066797280573,
1.79559541056502312623375820703, 2.90096053956357091792742845079, 4.28966781157781584591721105394, 5.30555883808069839160666986422, 6.41472717290222871948251639673, 7.67377592409065442110591953211, 8.679219804503891984246434936299, 8.944376798819313685277971476512, 10.29393468955933388677188215315, 11.48305053238713418620812243888