| L(s) = 1 | + (−1.85 − 0.0748i)2-s + (−0.278 + 0.960i)3-s + (1.45 + 0.117i)4-s + (−1.06 − 0.404i)5-s + (0.588 − 1.76i)6-s + (0.332 + 0.781i)7-s + (1.00 + 0.122i)8-s + (−0.845 − 0.534i)9-s + (1.95 + 0.832i)10-s + (−2.33 − 3.69i)11-s + (−0.515 + 1.36i)12-s + (−0.346 + 3.58i)13-s + (−0.559 − 1.47i)14-s + (0.686 − 0.912i)15-s + (−4.73 − 0.768i)16-s + (−2.27 + 0.970i)17-s + ⋯ |
| L(s) = 1 | + (−1.31 − 0.0529i)2-s + (−0.160 + 0.554i)3-s + (0.725 + 0.0585i)4-s + (−0.477 − 0.181i)5-s + (0.240 − 0.719i)6-s + (0.125 + 0.295i)7-s + (0.355 + 0.0431i)8-s + (−0.281 − 0.178i)9-s + (0.617 + 0.263i)10-s + (−0.703 − 1.11i)11-s + (−0.148 + 0.392i)12-s + (−0.0962 + 0.995i)13-s + (−0.149 − 0.394i)14-s + (0.177 − 0.235i)15-s + (−1.18 − 0.192i)16-s + (−0.552 + 0.235i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.587 + 0.809i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.587 + 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.392287 - 0.199895i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.392287 - 0.199895i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.278 - 0.960i)T \) |
| 13 | \( 1 + (0.346 - 3.58i)T \) |
| good | 2 | \( 1 + (1.85 + 0.0748i)T + (1.99 + 0.160i)T^{2} \) |
| 5 | \( 1 + (1.06 + 0.404i)T + (3.74 + 3.31i)T^{2} \) |
| 7 | \( 1 + (-0.332 - 0.781i)T + (-4.84 + 5.04i)T^{2} \) |
| 11 | \( 1 + (2.33 + 3.69i)T + (-4.71 + 9.93i)T^{2} \) |
| 17 | \( 1 + (2.27 - 0.970i)T + (11.7 - 12.2i)T^{2} \) |
| 19 | \( 1 + (-4.47 + 2.58i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.0828 - 0.143i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.381 + 9.46i)T + (-28.9 - 2.33i)T^{2} \) |
| 31 | \( 1 + (-4.15 - 4.68i)T + (-3.73 + 30.7i)T^{2} \) |
| 37 | \( 1 + (-5.07 + 1.03i)T + (34.0 - 14.5i)T^{2} \) |
| 41 | \( 1 + (0.363 + 0.105i)T + (34.6 + 21.9i)T^{2} \) |
| 43 | \( 1 + (-0.404 + 1.98i)T + (-39.5 - 16.8i)T^{2} \) |
| 47 | \( 1 + (-10.1 + 7.02i)T + (16.6 - 43.9i)T^{2} \) |
| 53 | \( 1 + (-0.281 + 2.31i)T + (-51.4 - 12.6i)T^{2} \) |
| 59 | \( 1 + (0.332 + 2.04i)T + (-55.9 + 18.6i)T^{2} \) |
| 61 | \( 1 + (-2.02 + 1.52i)T + (16.9 - 58.5i)T^{2} \) |
| 67 | \( 1 + (1.00 + 12.4i)T + (-66.1 + 10.7i)T^{2} \) |
| 71 | \( 1 + (3.85 - 3.70i)T + (2.85 - 70.9i)T^{2} \) |
| 73 | \( 1 + (-6.99 + 13.3i)T + (-41.4 - 60.0i)T^{2} \) |
| 79 | \( 1 + (2.73 + 3.95i)T + (-28.0 + 73.8i)T^{2} \) |
| 83 | \( 1 + (3.19 - 12.9i)T + (-73.4 - 38.5i)T^{2} \) |
| 89 | \( 1 + (8.81 + 5.08i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-7.32 - 5.98i)T + (19.4 + 95.0i)T^{2} \) |
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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
−10.63297144134841062203122063469, −9.786426770365691363097686783021, −8.976285362819269743425327893560, −8.348124772581426236287752076767, −7.53620534736193685048448122514, −6.30837649321837284084622327817, −5.05749762573354702604310940808, −4.01152171197404058008760276695, −2.41305499735024544897522992816, −0.49147799969606164341228560174,
1.14915660349405633127949975200, 2.66562042548128203528936815679, 4.37116285508258432642585627222, 5.58762036757835367932831239257, 7.09814535007369963843743000564, 7.53601639689562297833786515061, 8.155250229332904790059616055959, 9.288913578632254934423368604425, 10.12871903235081843745517811396, 10.79362553672738822156943688772
