| L(s) = 1 | + 15.5·3-s − 85.3i·5-s + 511. i·7-s + 243·9-s + 1.21e3·11-s + 2.74e3i·13-s − 1.33e3i·15-s − 8.52e3·17-s − 1.04e4·19-s + 7.97e3i·21-s − 4.30e3i·23-s + 8.33e3·25-s + 3.78e3·27-s + 4.24e4i·29-s − 5.78e4i·31-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.683i·5-s + 1.49i·7-s + 0.333·9-s + 0.911·11-s + 1.24i·13-s − 0.394i·15-s − 1.73·17-s − 1.52·19-s + 0.861i·21-s − 0.353i·23-s + 0.533·25-s + 0.192·27-s + 1.74i·29-s − 1.94i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.4722730309\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4722730309\) |
| \(L(4)\) |
|
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 | \( 1 + 85.3iT - 1.56e4T^{2} \) |
| 7 | \( 1 - 511. iT - 1.17e5T^{2} \) |
| 11 | \( 1 - 1.21e3T + 1.77e6T^{2} \) |
| 13 | \( 1 - 2.74e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 + 8.52e3T + 2.41e7T^{2} \) |
| 19 | \( 1 + 1.04e4T + 4.70e7T^{2} \) |
| 23 | \( 1 + 4.30e3iT - 1.48e8T^{2} \) |
| 29 | \( 1 - 4.24e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 + 5.78e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 + 8.35e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + 7.01e4T + 4.75e9T^{2} \) |
| 43 | \( 1 + 2.87e4T + 6.32e9T^{2} \) |
| 47 | \( 1 - 5.87e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 + 1.11e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 + 1.18e5T + 4.21e10T^{2} \) |
| 61 | \( 1 + 5.31e4iT - 5.15e10T^{2} \) |
| 67 | \( 1 - 4.21e4T + 9.04e10T^{2} \) |
| 71 | \( 1 - 3.51e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 9.12e4T + 1.51e11T^{2} \) |
| 79 | \( 1 + 7.75e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + 6.69e5T + 3.26e11T^{2} \) |
| 89 | \( 1 + 5.04e5T + 4.96e11T^{2} \) |
| 97 | \( 1 - 1.03e6T + 8.32e11T^{2} \) |
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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
−10.92858136941506541991316480994, −9.371147916841605453038931011106, −8.877371841534486144091117849497, −8.549010749155212011508228561082, −6.90167464497627094307635636882, −6.16110396086895832775319224905, −4.79517080203340849533309791313, −4.01015984861232240785358764736, −2.38105424716477628617959250810, −1.75146378184646070197170350343,
0.088891262865096915654709499252, 1.42473800533370296503052305837, 2.79900981900025898764319447287, 3.82831687435601877280185219003, 4.64444664602487873550310595151, 6.51738810286350384888740159480, 6.87899665602702434787022784358, 8.044564519651249868931568363942, 8.836203128996826016180951313921, 10.16772310427817493538698238761
