| L(s) = 1 | − 329.·3-s − 1.22e4·5-s − 8.71e4·7-s − 6.83e4·9-s + 3.43e5·11-s − 7.74e5·13-s + 4.02e6·15-s − 2.60e6·17-s − 1.33e7·19-s + 2.87e7·21-s − 4.56e7·23-s + 1.00e8·25-s + 8.09e7·27-s − 1.15e8·29-s − 3.59e7·31-s − 1.13e8·33-s + 1.06e9·35-s + 3.36e8·37-s + 2.55e8·39-s + 5.13e8·41-s + 4.70e8·43-s + 8.35e8·45-s + 2.11e9·47-s + 5.62e9·49-s + 8.59e8·51-s − 4.25e9·53-s − 4.19e9·55-s + ⋯ |
| L(s) = 1 | − 0.783·3-s − 1.74·5-s − 1.96·7-s − 0.385·9-s + 0.643·11-s − 0.578·13-s + 1.36·15-s − 0.445·17-s − 1.23·19-s + 1.53·21-s − 1.47·23-s + 2.05·25-s + 1.08·27-s − 1.04·29-s − 0.225·31-s − 0.504·33-s + 3.42·35-s + 0.798·37-s + 0.453·39-s + 0.692·41-s + 0.487·43-s + 0.674·45-s + 1.34·47-s + 2.84·49-s + 0.349·51-s − 1.39·53-s − 1.12·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + 329.T + 1.77e5T^{2} \) |
| 5 | \( 1 + 1.22e4T + 4.88e7T^{2} \) |
| 7 | \( 1 + 8.71e4T + 1.97e9T^{2} \) |
| 11 | \( 1 - 3.43e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 7.74e5T + 1.79e12T^{2} \) |
| 17 | \( 1 + 2.60e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 1.33e7T + 1.16e14T^{2} \) |
| 23 | \( 1 + 4.56e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 1.15e8T + 1.22e16T^{2} \) |
| 31 | \( 1 + 3.59e7T + 2.54e16T^{2} \) |
| 37 | \( 1 - 3.36e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 5.13e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 4.70e8T + 9.29e17T^{2} \) |
| 47 | \( 1 - 2.11e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 4.25e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + 3.15e8T + 3.01e19T^{2} \) |
| 61 | \( 1 + 4.85e8T + 4.35e19T^{2} \) |
| 67 | \( 1 + 4.92e9T + 1.22e20T^{2} \) |
| 71 | \( 1 - 2.43e10T + 2.31e20T^{2} \) |
| 73 | \( 1 - 1.87e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + 2.54e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + 4.47e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 7.21e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 3.03e9T + 7.15e21T^{2} \) |
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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
−9.626453697772656212300832050738, −8.666521554647632884672952890837, −7.50832516818348302254873783486, −6.59718250408551469537559113821, −5.90508167937851004028900969296, −4.30398894326298705776324897931, −3.71386436931235298415144859992, −2.59735018216116972080323646931, −0.50792440704267187784165607453, 0,
0.50792440704267187784165607453, 2.59735018216116972080323646931, 3.71386436931235298415144859992, 4.30398894326298705776324897931, 5.90508167937851004028900969296, 6.59718250408551469537559113821, 7.50832516818348302254873783486, 8.666521554647632884672952890837, 9.626453697772656212300832050738
