| L(s) = 1 | − 1.73·2-s + 0.999·4-s + 1.41i·5-s + 2.44i·7-s + 1.73·8-s − 2.44i·10-s + (1.73 + 2.82i)11-s + 4.89i·13-s − 4.24i·14-s − 5·16-s − 7.34i·19-s + 1.41i·20-s + (−2.99 − 4.89i)22-s − 2.82i·23-s + 2.99·25-s − 8.48i·26-s + ⋯ |
| L(s) = 1 | − 1.22·2-s + 0.499·4-s + 0.632i·5-s + 0.925i·7-s + 0.612·8-s − 0.774i·10-s + (0.522 + 0.852i)11-s + 1.35i·13-s − 1.13i·14-s − 1.25·16-s − 1.68i·19-s + 0.316i·20-s + (−0.639 − 1.04i)22-s − 0.589i·23-s + 0.599·25-s − 1.66i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.394 - 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.394 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.444946 + 0.293090i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.444946 + 0.293090i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + (-1.73 - 2.82i)T \) |
| good | 2 | \( 1 + 1.73T + 2T^{2} \) |
| 5 | \( 1 - 1.41iT - 5T^{2} \) |
| 7 | \( 1 - 2.44iT - 7T^{2} \) |
| 13 | \( 1 - 4.89iT - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 7.34iT - 19T^{2} \) |
| 23 | \( 1 + 2.82iT - 23T^{2} \) |
| 29 | \( 1 + 6.92T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 8T + 37T^{2} \) |
| 41 | \( 1 - 6.92T + 41T^{2} \) |
| 43 | \( 1 - 2.44iT - 43T^{2} \) |
| 47 | \( 1 + 2.82iT - 47T^{2} \) |
| 53 | \( 1 - 9.89iT - 53T^{2} \) |
| 59 | \( 1 + 11.3iT - 59T^{2} \) |
| 61 | \( 1 + 4.89iT - 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 2.82iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 12.2iT - 79T^{2} \) |
| 83 | \( 1 - 13.8T + 83T^{2} \) |
| 89 | \( 1 + 7.07iT - 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−14.32168190506878386511217532287, −12.95867149640898152762588436825, −11.60951141270858407331611013609, −10.81890712287236672964724032104, −9.293513445353055567601579767448, −9.121893942793375276437921306793, −7.47898061501936816669013541698, −6.55510790933723557155654578383, −4.56921065867110073474598274548, −2.21566271073912486023524403756,
1.03600880771872794815330881490, 3.88675360045157456971255696593, 5.67625452116918334045905826677, 7.46487163945053998074752161708, 8.231727043870037728492129332356, 9.324434967429838102217158277205, 10.32622616838747214066777289003, 11.17186905328940756867896011195, 12.71138264788252203224822483288, 13.60771438649453582831865666787
