L(s) = 1 | + (1.10 + 0.805i)2-s + (−1.36 + 0.993i)3-s + (−0.0371 − 0.114i)4-s − 0.800·5-s − 2.31·6-s + (−1.47 − 4.54i)7-s + (0.898 − 2.76i)8-s + (−0.0450 + 0.138i)9-s + (−0.887 − 0.645i)10-s + (−0.368 − 1.13i)11-s + (0.164 + 0.119i)12-s + (0.809 − 0.587i)13-s + (2.02 − 6.22i)14-s + (1.09 − 0.794i)15-s + (3.03 − 2.20i)16-s + (0.638 − 1.96i)17-s + ⋯ |
L(s) = 1 | + (0.784 + 0.569i)2-s + (−0.789 + 0.573i)3-s + (−0.0185 − 0.0571i)4-s − 0.357·5-s − 0.945·6-s + (−0.557 − 1.71i)7-s + (0.317 − 0.977i)8-s + (−0.0150 + 0.0462i)9-s + (−0.280 − 0.203i)10-s + (−0.111 − 0.342i)11-s + (0.0473 + 0.0344i)12-s + (0.224 − 0.163i)13-s + (0.540 − 1.66i)14-s + (0.282 − 0.205i)15-s + (0.757 − 0.550i)16-s + (0.154 − 0.476i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.463 + 0.886i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.463 + 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.886009 - 0.536468i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.886009 - 0.536468i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 + (-0.809 + 0.587i)T \) |
| 31 | \( 1 + (-2.65 - 4.89i)T \) |
good | 2 | \( 1 + (-1.10 - 0.805i)T + (0.618 + 1.90i)T^{2} \) |
| 3 | \( 1 + (1.36 - 0.993i)T + (0.927 - 2.85i)T^{2} \) |
| 5 | \( 1 + 0.800T + 5T^{2} \) |
| 7 | \( 1 + (1.47 + 4.54i)T + (-5.66 + 4.11i)T^{2} \) |
| 11 | \( 1 + (0.368 + 1.13i)T + (-8.89 + 6.46i)T^{2} \) |
| 17 | \( 1 + (-0.638 + 1.96i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (2.05 + 1.49i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-0.203 + 0.625i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (-1.84 - 1.34i)T + (8.96 + 27.5i)T^{2} \) |
| 37 | \( 1 - 6.15T + 37T^{2} \) |
| 41 | \( 1 + (-1.08 - 0.787i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (7.02 + 5.10i)T + (13.2 + 40.8i)T^{2} \) |
| 47 | \( 1 + (-2.38 + 1.73i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (-3.82 + 11.7i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (3.60 - 2.61i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + 5.00T + 61T^{2} \) |
| 67 | \( 1 + 2.60T + 67T^{2} \) |
| 71 | \( 1 + (-3.13 + 9.65i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-4.64 - 14.2i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-1.46 + 4.51i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-10.2 - 7.46i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (-1.50 - 4.63i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-0.345 - 1.06i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−10.89350536628900384634413218880, −10.39846611329503748919902877385, −9.647134850415886053826335561465, −8.025325300632780688016062709782, −7.00867225177514213099456925986, −6.27666959159652164513837901422, −5.16075330625907451306386423304, −4.36477697025811354711314617280, −3.53511777309932375721968391675, −0.57738369207005665251701728248,
2.08317343806396547487346648423, 3.25129273362840546256230922522, 4.52305430779626129388950476165, 5.79540630760736548349508968424, 6.19511039891071711532402895432, 7.71257246455794119984620180891, 8.637818914311233100512226137356, 9.658626274900634714692930669973, 11.05610241341829023732420643674, 11.79578947129971337354089033597