| L(s) = 1 | − 15.5·3-s − 196i·5-s − 270. i·7-s + 243·9-s − 1.72e3·11-s − 2.88e3i·13-s + 3.05e3i·15-s − 2.89e3·17-s − 9.78e3·19-s + 4.21e3i·21-s − 1.62e3i·23-s − 2.27e4·25-s − 3.78e3·27-s − 1.45e4i·29-s − 6.87e3i·31-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.56i·5-s − 0.787i·7-s + 0.333·9-s − 1.29·11-s − 1.31i·13-s + 0.905i·15-s − 0.589·17-s − 1.42·19-s + 0.454i·21-s − 0.133i·23-s − 1.45·25-s − 0.192·27-s − 0.598i·29-s − 0.230i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.5138333980\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5138333980\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 15.5T \) |
| good | 5 | \( 1 + 196iT - 1.56e4T^{2} \) |
| 7 | \( 1 + 270. iT - 1.17e5T^{2} \) |
| 11 | \( 1 + 1.72e3T + 1.77e6T^{2} \) |
| 13 | \( 1 + 2.88e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 + 2.89e3T + 2.41e7T^{2} \) |
| 19 | \( 1 + 9.78e3T + 4.70e7T^{2} \) |
| 23 | \( 1 + 1.62e3iT - 1.48e8T^{2} \) |
| 29 | \( 1 + 1.45e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 + 6.87e3iT - 8.87e8T^{2} \) |
| 37 | \( 1 - 1.21e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 - 1.87e4T + 4.75e9T^{2} \) |
| 43 | \( 1 + 1.35e5T + 6.32e9T^{2} \) |
| 47 | \( 1 + 1.51e5iT - 1.07e10T^{2} \) |
| 53 | \( 1 + 1.44e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 - 3.33e5T + 4.21e10T^{2} \) |
| 61 | \( 1 + 3.33e5iT - 5.15e10T^{2} \) |
| 67 | \( 1 - 3.19e5T + 9.04e10T^{2} \) |
| 71 | \( 1 - 4.20e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 6.28e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + 2.91e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 - 3.57e5T + 3.26e11T^{2} \) |
| 89 | \( 1 + 1.37e6T + 4.96e11T^{2} \) |
| 97 | \( 1 - 4.89e5T + 8.32e11T^{2} \) |
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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
−9.870580654210110300713233247290, −8.436227877240400131231947714527, −8.065643998141751936253290183161, −6.74290217924314945939109168217, −5.46593869072315604433969721156, −4.90992198146446574304214192404, −3.88583530045502840240003720090, −2.13810960387850130860096271744, −0.67177025891582614369999627356, −0.18294508809648738156515284914,
2.01966427262219881908699252172, 2.76460912106516714180348634485, 4.17506872885539156188605915225, 5.40609899904537487430425394022, 6.45794921491943076880096618464, 6.96611170853890438030759441486, 8.175272441759025628786344469077, 9.301903354722870704395298366267, 10.43784399695358003166285418732, 10.91069007167381046721120156024
