| L(s) = 1 | − 3.01·2-s − 4.83·3-s + 5.10·4-s + 14.5·6-s − 6.15i·7-s − 3.33·8-s + 14.3·9-s + 5.09·11-s − 24.6·12-s + 4.20·13-s + 18.5i·14-s − 10.3·16-s + 1.29i·17-s − 43.2·18-s + (−18.3 − 4.84i)19-s + ⋯ |
| L(s) = 1 | − 1.50·2-s − 1.61·3-s + 1.27·4-s + 2.43·6-s − 0.879i·7-s − 0.417·8-s + 1.59·9-s + 0.463·11-s − 2.05·12-s + 0.323·13-s + 1.32i·14-s − 0.647·16-s + 0.0764i·17-s − 2.40·18-s + (−0.966 − 0.255i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.750 + 0.660i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.750 + 0.660i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3508762333\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3508762333\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 + (18.3 + 4.84i)T \) |
| good | 2 | \( 1 + 3.01T + 4T^{2} \) |
| 3 | \( 1 + 4.83T + 9T^{2} \) |
| 7 | \( 1 + 6.15iT - 49T^{2} \) |
| 11 | \( 1 - 5.09T + 121T^{2} \) |
| 13 | \( 1 - 4.20T + 169T^{2} \) |
| 17 | \( 1 - 1.29iT - 289T^{2} \) |
| 23 | \( 1 - 37.1iT - 529T^{2} \) |
| 29 | \( 1 + 42.9iT - 841T^{2} \) |
| 31 | \( 1 - 52.1iT - 961T^{2} \) |
| 37 | \( 1 + 41.8T + 1.36e3T^{2} \) |
| 41 | \( 1 - 7.13iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 59.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 57.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 61.6T + 2.80e3T^{2} \) |
| 59 | \( 1 + 54.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 38.4T + 3.72e3T^{2} \) |
| 67 | \( 1 - 44.7T + 4.48e3T^{2} \) |
| 71 | \( 1 + 13.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 134. iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 128. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 23.2iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 21.0iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 75.9T + 9.40e3T^{2} \) |
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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
−10.54167659698259636319955054998, −10.07430346922916962528428906094, −9.049255197388677303566553535483, −7.957145426798956311377020757087, −6.97173556245849704815402774367, −6.42813833534070899329859007488, −5.16213656692017931824137462012, −3.97389404776464289830324969856, −1.58243407799278295818964478334, −0.52130684140304332527521119762,
0.70222830838186206266153334852, 2.11846212921434782221262007215, 4.33742524458455410211943152742, 5.56020919606831467851010854496, 6.41594745850290795059742922834, 7.14936189444270612331779101631, 8.496591966120723493658122658737, 9.034908129088112428892258223841, 10.25457579674570045723347405140, 10.67510979224379114704502805609
