L(s) = 1 | + (1.64 + 1.64i)2-s + (1.43 + 1.43i)3-s + 3.44i·4-s + (1.13 + 1.92i)5-s + 4.75i·6-s + (−3.25 − 3.25i)7-s + (−2.37 + 2.37i)8-s + 1.14i·9-s + (−1.30 + 5.05i)10-s + (−4.95 + 4.95i)12-s + (−1.73 + 1.73i)13-s − 10.7i·14-s + (−1.13 + 4.40i)15-s − 0.961·16-s + (1.32 + 1.32i)17-s + (−1.89 + 1.89i)18-s + ⋯ |
L(s) = 1 | + (1.16 + 1.16i)2-s + (0.831 + 0.831i)3-s + 1.72i·4-s + (0.507 + 0.861i)5-s + 1.93i·6-s + (−1.23 − 1.23i)7-s + (−0.840 + 0.840i)8-s + 0.382i·9-s + (−0.412 + 1.59i)10-s + (−1.43 + 1.43i)12-s + (−0.480 + 0.480i)13-s − 2.87i·14-s + (−0.294 + 1.13i)15-s − 0.240·16-s + (0.322 + 0.322i)17-s + (−0.445 + 0.445i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.841 - 0.540i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.841 - 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.939440 + 3.20368i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.939440 + 3.20368i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-1.13 - 1.92i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-1.64 - 1.64i)T + 2iT^{2} \) |
| 3 | \( 1 + (-1.43 - 1.43i)T + 3iT^{2} \) |
| 7 | \( 1 + (3.25 + 3.25i)T + 7iT^{2} \) |
| 13 | \( 1 + (1.73 - 1.73i)T - 13iT^{2} \) |
| 17 | \( 1 + (-1.32 - 1.32i)T + 17iT^{2} \) |
| 19 | \( 1 - 1.71T + 19T^{2} \) |
| 23 | \( 1 + (-0.313 - 0.313i)T + 23iT^{2} \) |
| 29 | \( 1 - 2.83T + 29T^{2} \) |
| 31 | \( 1 - 2.34T + 31T^{2} \) |
| 37 | \( 1 + (-1.29 + 1.29i)T - 37iT^{2} \) |
| 41 | \( 1 + 7.57iT - 41T^{2} \) |
| 43 | \( 1 + (-4.63 + 4.63i)T - 43iT^{2} \) |
| 47 | \( 1 + (-1.55 + 1.55i)T - 47iT^{2} \) |
| 53 | \( 1 + (9.45 + 9.45i)T + 53iT^{2} \) |
| 59 | \( 1 + 4.14iT - 59T^{2} \) |
| 61 | \( 1 + 2.45iT - 61T^{2} \) |
| 67 | \( 1 + (-8.24 + 8.24i)T - 67iT^{2} \) |
| 71 | \( 1 + 9.33T + 71T^{2} \) |
| 73 | \( 1 + (9.98 - 9.98i)T - 73iT^{2} \) |
| 79 | \( 1 + 2.12T + 79T^{2} \) |
| 83 | \( 1 + (10.5 - 10.5i)T - 83iT^{2} \) |
| 89 | \( 1 - 4.75iT - 89T^{2} \) |
| 97 | \( 1 + (9.77 - 9.77i)T - 97iT^{2} \) |
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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
−10.73679690278479964282628366476, −9.959264353606630957906042008851, −9.431174223656227110216237444240, −8.071809540056816549530743612937, −7.01869854596570614248237499010, −6.67125913648234531643629344740, −5.59155620357481277306162983143, −4.28334386343161059281130788523, −3.63695378966488510533794348912, −2.88553261189461659511390546496,
1.37083633642121902815044215265, 2.71029119744681962481246568681, 2.91219611671402194997920165233, 4.57257073366137865127607899768, 5.55259409432498595882425490662, 6.27764206468945071196330439922, 7.72880958472696191566627770323, 8.777092535438074705162643770535, 9.552261222113207691993414213574, 10.23995120953294771196873625517