| L(s) = 1 | + (1.89 − 1.89i)3-s + (0.372 + 0.372i)5-s + i·7-s − 4.21i·9-s + (0.158 + 0.158i)11-s + (−1.48 + 1.48i)13-s + 1.41·15-s + 3.79·17-s + (2.94 − 2.94i)19-s + (1.89 + 1.89i)21-s − 1.95i·23-s − 4.72i·25-s + (−2.30 − 2.30i)27-s + (0.0304 − 0.0304i)29-s + 9.16·31-s + ⋯ |
| L(s) = 1 | + (1.09 − 1.09i)3-s + (0.166 + 0.166i)5-s + 0.377i·7-s − 1.40i·9-s + (0.0479 + 0.0479i)11-s + (−0.411 + 0.411i)13-s + 0.365·15-s + 0.921·17-s + (0.675 − 0.675i)19-s + (0.414 + 0.414i)21-s − 0.408i·23-s − 0.944i·25-s + (−0.442 − 0.442i)27-s + (0.00566 − 0.00566i)29-s + 1.64·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.701679877\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.701679877\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - iT \) |
| good | 3 | \( 1 + (-1.89 + 1.89i)T - 3iT^{2} \) |
| 5 | \( 1 + (-0.372 - 0.372i)T + 5iT^{2} \) |
| 11 | \( 1 + (-0.158 - 0.158i)T + 11iT^{2} \) |
| 13 | \( 1 + (1.48 - 1.48i)T - 13iT^{2} \) |
| 17 | \( 1 - 3.79T + 17T^{2} \) |
| 19 | \( 1 + (-2.94 + 2.94i)T - 19iT^{2} \) |
| 23 | \( 1 + 1.95iT - 23T^{2} \) |
| 29 | \( 1 + (-0.0304 + 0.0304i)T - 29iT^{2} \) |
| 31 | \( 1 - 9.16T + 31T^{2} \) |
| 37 | \( 1 + (5.78 + 5.78i)T + 37iT^{2} \) |
| 41 | \( 1 + 10.9iT - 41T^{2} \) |
| 43 | \( 1 + (-2.62 - 2.62i)T + 43iT^{2} \) |
| 47 | \( 1 - 7.79T + 47T^{2} \) |
| 53 | \( 1 + (-2.21 - 2.21i)T + 53iT^{2} \) |
| 59 | \( 1 + (9.17 + 9.17i)T + 59iT^{2} \) |
| 61 | \( 1 + (-4.97 + 4.97i)T - 61iT^{2} \) |
| 67 | \( 1 + (4.42 - 4.42i)T - 67iT^{2} \) |
| 71 | \( 1 - 14.5iT - 71T^{2} \) |
| 73 | \( 1 - 12.0iT - 73T^{2} \) |
| 79 | \( 1 + 11.4T + 79T^{2} \) |
| 83 | \( 1 + (-0.123 + 0.123i)T - 83iT^{2} \) |
| 89 | \( 1 - 6.79iT - 89T^{2} \) |
| 97 | \( 1 - 8.80T + 97T^{2} \) |
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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
−8.915955691855526601822251366597, −8.372385774740991462863160430340, −7.50499823912943918555002834330, −6.99428949022077510889563990187, −6.13637705253621582963615889427, −5.12028116583535190369313454449, −3.89811874602735268933299510672, −2.76913283530080844137354631110, −2.27200687600309921890928734751, −0.992622387298509148495470203464,
1.38137162176334237125800506318, 2.90363839003348271449949906289, 3.39015418082005125499114132190, 4.39787309582370521664232966253, 5.13496681976117150465853055387, 6.08494045070170406362471695570, 7.41709608349535868336054438049, 7.930729355144290354880890843716, 8.763922408973798249060881606752, 9.482505538922961437709273907761
