| L(s) = 1 | − 1.41·3-s − 2.44i·5-s + (2 − 1.73i)7-s − 0.999·9-s + 4.89i·11-s − 2.44i·13-s + 3.46i·15-s + 6.92i·17-s − 4.24·19-s + (−2.82 + 2.44i)21-s + 6.92i·23-s − 0.999·25-s + 5.65·27-s − 5.65·29-s + 4·31-s + ⋯ |
| L(s) = 1 | − 0.816·3-s − 1.09i·5-s + (0.755 − 0.654i)7-s − 0.333·9-s + 1.47i·11-s − 0.679i·13-s + 0.894i·15-s + 1.68i·17-s − 0.973·19-s + (−0.617 + 0.534i)21-s + 1.44i·23-s − 0.199·25-s + 1.08·27-s − 1.05·29-s + 0.718·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.017184683\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.017184683\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (-2 + 1.73i)T \) |
| good | 3 | \( 1 + 1.41T + 3T^{2} \) |
| 5 | \( 1 + 2.44iT - 5T^{2} \) |
| 11 | \( 1 - 4.89iT - 11T^{2} \) |
| 13 | \( 1 + 2.44iT - 13T^{2} \) |
| 17 | \( 1 - 6.92iT - 17T^{2} \) |
| 19 | \( 1 + 4.24T + 19T^{2} \) |
| 23 | \( 1 - 6.92iT - 23T^{2} \) |
| 29 | \( 1 + 5.65T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - 4.89iT - 43T^{2} \) |
| 47 | \( 1 - 12T + 47T^{2} \) |
| 53 | \( 1 + 5.65T + 53T^{2} \) |
| 59 | \( 1 - 9.89T + 59T^{2} \) |
| 61 | \( 1 + 7.34iT - 61T^{2} \) |
| 67 | \( 1 - 4.89iT - 67T^{2} \) |
| 71 | \( 1 - 3.46iT - 71T^{2} \) |
| 73 | \( 1 + 6.92iT - 73T^{2} \) |
| 79 | \( 1 - 10.3iT - 79T^{2} \) |
| 83 | \( 1 + 1.41T + 83T^{2} \) |
| 89 | \( 1 - 6.92iT - 89T^{2} \) |
| 97 | \( 1 - 6.92iT - 97T^{2} \) |
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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
−9.400073904230028586183227173296, −8.429746513936627399537858988526, −7.918306052177600550977658484151, −7.00448369945804690558429580636, −5.95809373960077762526223619878, −5.26910393122819483417100100901, −4.56131907116604982442377514833, −3.83713970382752904582177295514, −2.02753629813505471422964235063, −1.07563535258942715897234307987,
0.49313560819150950505062716016, 2.34685926912015343251861729281, 3.00863368525023218077480816651, 4.35047628305568094916447832867, 5.28890331841303674749806810449, 6.02391074156949579476707711229, 6.61924582362905393053726837349, 7.47068807922424570610880014776, 8.635291274477603315006059421978, 8.909448806247841792914696067942
