| L(s) = 1 | − 13.0·3-s + 23.8·5-s + 161.·7-s − 71.4·9-s − 675.·11-s − 631.·13-s − 313.·15-s − 1.09e3·17-s − 2.77e3·19-s − 2.12e3·21-s − 1.12e3·23-s − 2.55e3·25-s + 4.11e3·27-s − 613.·29-s − 6.88e3·31-s + 8.84e3·33-s + 3.87e3·35-s + 7.54e3·37-s + 8.27e3·39-s + 9.83e3·41-s + 5.25e3·43-s − 1.70e3·45-s − 1.40e4·47-s + 9.41e3·49-s + 1.43e4·51-s + 8.79e3·53-s − 1.61e4·55-s + ⋯ |
| L(s) = 1 | − 0.840·3-s + 0.427·5-s + 1.24·7-s − 0.293·9-s − 1.68·11-s − 1.03·13-s − 0.359·15-s − 0.918·17-s − 1.76·19-s − 1.04·21-s − 0.443·23-s − 0.817·25-s + 1.08·27-s − 0.135·29-s − 1.28·31-s + 1.41·33-s + 0.534·35-s + 0.906·37-s + 0.871·39-s + 0.913·41-s + 0.433·43-s − 0.125·45-s − 0.928·47-s + 0.560·49-s + 0.771·51-s + 0.430·53-s − 0.719·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1028 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.5139087011\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5139087011\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 257 | \( 1 + 6.60e4T \) |
| good | 3 | \( 1 + 13.0T + 243T^{2} \) |
| 5 | \( 1 - 23.8T + 3.12e3T^{2} \) |
| 7 | \( 1 - 161.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 675.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 631.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.09e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.77e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.12e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 613.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.88e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 7.54e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 9.83e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 5.25e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.40e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 8.79e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.48e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.42e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.40e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.99e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.93e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.01e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.21e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 2.34e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 4.45e4T + 8.58e9T^{2} \) |
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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
−9.231352967236110083821037566583, −8.184936175012806081044462026699, −7.70263535796507079915003643044, −6.51519092943657000423922750987, −5.63065119892573384637940098709, −5.03004157342922720864167635859, −4.29714696726293955296130884057, −2.48491183618505511805271681462, −1.98747750241281218193949422297, −0.30267259287847879219039794125,
0.30267259287847879219039794125, 1.98747750241281218193949422297, 2.48491183618505511805271681462, 4.29714696726293955296130884057, 5.03004157342922720864167635859, 5.63065119892573384637940098709, 6.51519092943657000423922750987, 7.70263535796507079915003643044, 8.184936175012806081044462026699, 9.231352967236110083821037566583
