| L(s) = 1 | + 24.5i·2-s + 131. i·3-s + 423.·4-s + 1.06e3i·5-s − 3.22e3·6-s − 2.31e4i·7-s + 3.54e4i·8-s + 4.17e4·9-s − 2.61e4·10-s + 1.58e5·11-s + 5.57e4i·12-s + 1.67e5·13-s + 5.68e5·14-s − 1.40e5·15-s − 4.35e5·16-s + 5.20e5·17-s + ⋯ |
| L(s) = 1 | + 0.765i·2-s + 0.541i·3-s + 0.413·4-s + 0.341i·5-s − 0.414·6-s − 1.37i·7-s + 1.08i·8-s + 0.706·9-s − 0.261·10-s + 0.986·11-s + 0.224i·12-s + 0.451·13-s + 1.05·14-s − 0.184·15-s − 0.415·16-s + 0.366·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.186 - 0.982i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.186 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(2.12729 + 1.76209i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.12729 + 1.76209i\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (2.73e7 - 1.44e8i)T \) |
| good | 2 | \( 1 - 24.5iT - 1.02e3T^{2} \) |
| 3 | \( 1 - 131. iT - 5.90e4T^{2} \) |
| 5 | \( 1 - 1.06e3iT - 9.76e6T^{2} \) |
| 7 | \( 1 + 2.31e4iT - 2.82e8T^{2} \) |
| 11 | \( 1 - 1.58e5T + 2.59e10T^{2} \) |
| 13 | \( 1 - 1.67e5T + 1.37e11T^{2} \) |
| 17 | \( 1 - 5.20e5T + 2.01e12T^{2} \) |
| 19 | \( 1 + 3.58e6iT - 6.13e12T^{2} \) |
| 23 | \( 1 - 5.91e6T + 4.14e13T^{2} \) |
| 29 | \( 1 - 2.07e7iT - 4.20e14T^{2} \) |
| 31 | \( 1 + 5.10e7T + 8.19e14T^{2} \) |
| 37 | \( 1 - 2.80e7iT - 4.80e15T^{2} \) |
| 41 | \( 1 + 3.53e7T + 1.34e16T^{2} \) |
| 47 | \( 1 + 1.53e8T + 5.25e16T^{2} \) |
| 53 | \( 1 - 6.12e8T + 1.74e17T^{2} \) |
| 59 | \( 1 - 1.45e8T + 5.11e17T^{2} \) |
| 61 | \( 1 - 2.72e8iT - 7.13e17T^{2} \) |
| 67 | \( 1 - 1.22e9T + 1.82e18T^{2} \) |
| 71 | \( 1 + 1.66e9iT - 3.25e18T^{2} \) |
| 73 | \( 1 + 3.53e9iT - 4.29e18T^{2} \) |
| 79 | \( 1 + 5.02e7T + 9.46e18T^{2} \) |
| 83 | \( 1 + 7.32e9T + 1.55e19T^{2} \) |
| 89 | \( 1 - 5.46e9iT - 3.11e19T^{2} \) |
| 97 | \( 1 - 1.01e10T + 7.37e19T^{2} \) |
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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
−14.38404059953512578532156994794, −13.09784002774004934901400908694, −11.25299083586217774153905039453, −10.54394098308313634695885677699, −9.037939235164506680221271260530, −7.25644846968107565774751117855, −6.75124519787373655844056521557, −4.85982431431458501039200799408, −3.44562383419612444515603260613, −1.24155269505815859623675883538,
1.15458822181859401286521677534, 2.09440794233832027384025928971, 3.71620553852346632615821227453, 5.78927035252357106099610672096, 7.05636438657948698881455354262, 8.703162559095418572246353834508, 9.917845847939480312980509977253, 11.42114076652915771903770141574, 12.30800586495658097682767432498, 12.90565247518115629410736528150
