| L(s) = 1 | − 24.5i·3-s + 49i·7-s − 359.·9-s + 90.2·11-s + 14.4i·13-s + 407. i·17-s − 2.28e3·19-s + 1.20e3·21-s + 505. i·23-s + 2.85e3i·27-s + 3.16e3·29-s − 6.23e3·31-s − 2.21e3i·33-s + 5.38e3i·37-s + 354.·39-s + ⋯ |
| L(s) = 1 | − 1.57i·3-s + 0.377i·7-s − 1.47·9-s + 0.224·11-s + 0.0236i·13-s + 0.341i·17-s − 1.45·19-s + 0.595·21-s + 0.199i·23-s + 0.754i·27-s + 0.698·29-s − 1.16·31-s − 0.354i·33-s + 0.647i·37-s + 0.0373·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.654688363\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.654688363\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 49iT \) |
| good | 3 | \( 1 + 24.5iT - 243T^{2} \) |
| 11 | \( 1 - 90.2T + 1.61e5T^{2} \) |
| 13 | \( 1 - 14.4iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 407. iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 2.28e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 505. iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 3.16e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.23e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 5.38e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 1.17e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 5.82e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 7.34e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 1.49e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 4.71e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.28e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.88e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 5.85e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.59e3iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 7.91e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 7.14e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 7.71e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.15e4iT - 8.58e9T^{2} \) |
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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
−9.436951756833517207647645300585, −8.471140568143427615620752892675, −7.926298895913594964295023004044, −6.85487780310894976723328817599, −6.35452872887554010730296587048, −5.39038518321104377738542224561, −4.00893469577725371635916367787, −2.60631501476762742201909060390, −1.82375153957687338823949971767, −0.77230428127230157803668292140,
0.46066245388220388121517235430, 2.22978328907376025874699502392, 3.51771690953161198621399785381, 4.20926557938158378355747168224, 5.00130387518343635566408852654, 6.01067446598911200335893954307, 7.09060921472928119391075572202, 8.300874914641278507343318921864, 9.087424324715515881122258648356, 9.747993255194303500347523712151
