| L(s) = 1 | + (−11.9 − 20.6i)3-s + (−541. + 937. i)5-s + (5.62e3 + 2.95e3i)7-s + (9.55e3 − 1.65e4i)9-s + (3.74e4 + 6.48e4i)11-s − 5.00e4·13-s + 2.58e4·15-s + (−5.94e4 − 1.03e5i)17-s + (4.02e5 − 6.96e5i)19-s + (−6.09e3 − 1.51e5i)21-s + (1.34e5 − 2.32e5i)23-s + (3.90e5 + 6.76e5i)25-s − 9.25e5·27-s − 5.29e6·29-s + (2.73e6 + 4.73e6i)31-s + ⋯ |
| L(s) = 1 | + (−0.0849 − 0.147i)3-s + (−0.387 + 0.670i)5-s + (0.885 + 0.464i)7-s + (0.485 − 0.841i)9-s + (0.771 + 1.33i)11-s − 0.486·13-s + 0.131·15-s + (−0.172 − 0.299i)17-s + (0.707 − 1.22i)19-s + (−0.00683 − 0.169i)21-s + (0.100 − 0.173i)23-s + (0.200 + 0.346i)25-s − 0.335·27-s − 1.39·29-s + (0.531 + 0.920i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.407 - 0.913i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.407 - 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(2.130310895\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.130310895\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (-5.62e3 - 2.95e3i)T \) |
| good | 3 | \( 1 + (11.9 + 20.6i)T + (-9.84e3 + 1.70e4i)T^{2} \) |
| 5 | \( 1 + (541. - 937. i)T + (-9.76e5 - 1.69e6i)T^{2} \) |
| 11 | \( 1 + (-3.74e4 - 6.48e4i)T + (-1.17e9 + 2.04e9i)T^{2} \) |
| 13 | \( 1 + 5.00e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + (5.94e4 + 1.03e5i)T + (-5.92e10 + 1.02e11i)T^{2} \) |
| 19 | \( 1 + (-4.02e5 + 6.96e5i)T + (-1.61e11 - 2.79e11i)T^{2} \) |
| 23 | \( 1 + (-1.34e5 + 2.32e5i)T + (-9.00e11 - 1.55e12i)T^{2} \) |
| 29 | \( 1 + 5.29e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + (-2.73e6 - 4.73e6i)T + (-1.32e13 + 2.28e13i)T^{2} \) |
| 37 | \( 1 + (1.55e6 - 2.69e6i)T + (-6.49e13 - 1.12e14i)T^{2} \) |
| 41 | \( 1 - 2.07e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 4.76e6T + 5.02e14T^{2} \) |
| 47 | \( 1 + (-1.50e7 + 2.61e7i)T + (-5.59e14 - 9.69e14i)T^{2} \) |
| 53 | \( 1 + (2.85e7 + 4.95e7i)T + (-1.64e15 + 2.85e15i)T^{2} \) |
| 59 | \( 1 + (-9.18e7 - 1.59e8i)T + (-4.33e15 + 7.50e15i)T^{2} \) |
| 61 | \( 1 + (2.65e7 - 4.59e7i)T + (-5.84e15 - 1.01e16i)T^{2} \) |
| 67 | \( 1 + (-8.67e7 - 1.50e8i)T + (-1.36e16 + 2.35e16i)T^{2} \) |
| 71 | \( 1 - 1.68e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + (-2.32e8 - 4.02e8i)T + (-2.94e16 + 5.09e16i)T^{2} \) |
| 79 | \( 1 + (2.06e8 - 3.58e8i)T + (-5.99e16 - 1.03e17i)T^{2} \) |
| 83 | \( 1 + 3.50e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + (2.58e8 - 4.48e8i)T + (-1.75e17 - 3.03e17i)T^{2} \) |
| 97 | \( 1 - 1.32e9T + 7.60e17T^{2} \) |
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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
−11.89460390820655880737652192955, −11.25700472053457959063294184368, −9.833434208580710282020675570040, −8.945191870607163231037733046631, −7.33006512942034888677848208193, −6.87708209873570488483489166802, −5.15772521116385414519297978378, −4.00611563073335929938596670461, −2.47599342828489948513873965213, −1.13687397869607575557061428364,
0.64076381775357206044423029166, 1.76151803422820221911403792883, 3.73143600233812282524124567387, 4.70380732397408470933896553083, 5.86900526832538538065921821217, 7.58291075050274620999833665428, 8.236562620835905259851880890317, 9.492586278943686306767544510143, 10.79824205848597675208919288376, 11.51821081271916970138049573634
