L(s) = 1 | + (−0.608 − 0.793i)2-s + (−0.258 + 0.965i)4-s + (1.05 + 0.357i)5-s + (0.923 − 0.382i)8-s + (0.991 + 0.130i)9-s + (−0.357 − 1.05i)10-s + (1 + i)13-s + (−0.866 − 0.499i)16-s + (−0.130 − 0.991i)17-s + (−0.499 − 0.866i)18-s + (−0.617 + 0.923i)20-s + (0.186 + 0.142i)25-s + (0.184 − 1.40i)26-s + (−0.216 − 1.08i)29-s + (0.130 + 0.991i)32-s + ⋯ |
L(s) = 1 | + (−0.608 − 0.793i)2-s + (−0.258 + 0.965i)4-s + (1.05 + 0.357i)5-s + (0.923 − 0.382i)8-s + (0.991 + 0.130i)9-s + (−0.357 − 1.05i)10-s + (1 + i)13-s + (−0.866 − 0.499i)16-s + (−0.130 − 0.991i)17-s + (−0.499 − 0.866i)18-s + (−0.617 + 0.923i)20-s + (0.186 + 0.142i)25-s + (0.184 − 1.40i)26-s + (−0.216 − 1.08i)29-s + (0.130 + 0.991i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 + 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 + 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.260490341\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.260490341\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.608 + 0.793i)T \) |
| 7 | \( 1 \) |
| 17 | \( 1 + (0.130 + 0.991i)T \) |
good | 3 | \( 1 + (-0.991 - 0.130i)T^{2} \) |
| 5 | \( 1 + (-1.05 - 0.357i)T + (0.793 + 0.608i)T^{2} \) |
| 11 | \( 1 + (0.608 + 0.793i)T^{2} \) |
| 13 | \( 1 + (-1 - i)T + iT^{2} \) |
| 19 | \( 1 + (-0.258 + 0.965i)T^{2} \) |
| 23 | \( 1 + (0.991 - 0.130i)T^{2} \) |
| 29 | \( 1 + (0.216 + 1.08i)T + (-0.923 + 0.382i)T^{2} \) |
| 31 | \( 1 + (0.991 + 0.130i)T^{2} \) |
| 37 | \( 1 + (-1.49 + 0.735i)T + (0.608 - 0.793i)T^{2} \) |
| 41 | \( 1 + (0.0761 - 0.382i)T + (-0.923 - 0.382i)T^{2} \) |
| 43 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 47 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 53 | \( 1 + (1.83 - 0.241i)T + (0.965 - 0.258i)T^{2} \) |
| 59 | \( 1 + (0.258 + 0.965i)T^{2} \) |
| 61 | \( 1 + (1.29 - 1.47i)T + (-0.130 - 0.991i)T^{2} \) |
| 67 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.382 - 0.923i)T^{2} \) |
| 73 | \( 1 + (1.25 - 1.09i)T + (0.130 - 0.991i)T^{2} \) |
| 79 | \( 1 + (-0.991 + 0.130i)T^{2} \) |
| 83 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 89 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 97 | \( 1 + (-1.63 + 0.324i)T + (0.923 - 0.382i)T^{2} \) |
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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
−9.220277042375867753857532785913, −8.076528755421906125418161609008, −7.36920567138313603483294043605, −6.59635666399401870136614512860, −5.89684590354402991110130771020, −4.60525947405653588351217057143, −4.06412250101464644010068383183, −2.89000326733841961606683478413, −2.06778309368915818642433088642, −1.26242599526066894770155284304,
1.20835503897156078869665729075, 1.83811639308141362873532887781, 3.38809986507536881825822977876, 4.51876462746256053539642016659, 5.25267308233358990546222721439, 6.19442592356842810173246289385, 6.33038074695391050446788874403, 7.52391157952636980027985164993, 8.065966568746956959050157990284, 8.925911487940695155658090908969