| L(s) = 1 | + (−1.00 + 1.95i)2-s + (−0.109 − 0.998i)3-s + (−0.488 − 0.683i)4-s + (0.139 + 0.0654i)5-s + (2.06 + 0.789i)6-s + (−0.973 − 4.36i)7-s + (−6.87 + 1.01i)8-s + (7.80 − 1.73i)9-s + (−0.268 + 0.207i)10-s + (0.328 − 0.184i)11-s + (−0.628 + 0.563i)12-s + (−15.4 + 7.94i)13-s + (9.52 + 2.49i)14-s + (0.0499 − 0.146i)15-s + (6.02 − 17.6i)16-s + (−10.7 − 19.2i)17-s + ⋯ |
| L(s) = 1 | + (−0.502 + 0.978i)2-s + (−0.0366 − 0.332i)3-s + (−0.122 − 0.170i)4-s + (0.0279 + 0.0130i)5-s + (0.343 + 0.131i)6-s + (−0.139 − 0.624i)7-s + (−0.859 + 0.126i)8-s + (0.866 − 0.193i)9-s + (−0.0268 + 0.0207i)10-s + (0.0298 − 0.0167i)11-s + (−0.0523 + 0.0469i)12-s + (−1.18 + 0.610i)13-s + (0.680 + 0.177i)14-s + (0.00333 − 0.00977i)15-s + (0.376 − 1.10i)16-s + (−0.634 − 1.13i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.203 + 0.979i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.203 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.395452 - 0.321616i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.395452 - 0.321616i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 431 | \( 1 + (409. - 135. i)T \) |
| good | 2 | \( 1 + (1.00 - 1.95i)T + (-2.32 - 3.25i)T^{2} \) |
| 3 | \( 1 + (0.109 + 0.998i)T + (-8.78 + 1.95i)T^{2} \) |
| 5 | \( 1 + (-0.139 - 0.0654i)T + (15.9 + 19.2i)T^{2} \) |
| 7 | \( 1 + (0.973 + 4.36i)T + (-44.3 + 20.7i)T^{2} \) |
| 11 | \( 1 + (-0.328 + 0.184i)T + (63.0 - 103. i)T^{2} \) |
| 13 | \( 1 + (15.4 - 7.94i)T + (98.3 - 137. i)T^{2} \) |
| 17 | \( 1 + (10.7 + 19.2i)T + (-150. + 246. i)T^{2} \) |
| 19 | \( 1 + (6.11 - 2.86i)T + (230. - 277. i)T^{2} \) |
| 23 | \( 1 + (-4.13 - 4.28i)T + (-19.3 + 528. i)T^{2} \) |
| 29 | \( 1 + (15.7 - 22.0i)T + (-271. - 795. i)T^{2} \) |
| 31 | \( 1 + (4.38 + 11.4i)T + (-715. + 641. i)T^{2} \) |
| 37 | \( 1 + (42.7 + 11.1i)T + (1.19e3 + 670. i)T^{2} \) |
| 41 | \( 1 + (-20.0 + 58.7i)T + (-1.33e3 - 1.02e3i)T^{2} \) |
| 43 | \( 1 + (4.87 - 6.31i)T + (-467. - 1.78e3i)T^{2} \) |
| 47 | \( 1 + (-2.11 + 28.8i)T + (-2.18e3 - 321. i)T^{2} \) |
| 53 | \( 1 + (-23.4 + 1.71i)T + (2.77e3 - 408. i)T^{2} \) |
| 59 | \( 1 + (3.51 - 6.84i)T + (-2.02e3 - 2.83e3i)T^{2} \) |
| 61 | \( 1 + (6.56 + 15.4i)T + (-2.58e3 + 2.67e3i)T^{2} \) |
| 67 | \( 1 + (66.9 + 17.5i)T + (3.91e3 + 2.19e3i)T^{2} \) |
| 71 | \( 1 + (-7.50 + 102. i)T + (-4.98e3 - 733. i)T^{2} \) |
| 73 | \( 1 + (-22.8 - 34.4i)T + (-2.08e3 + 4.90e3i)T^{2} \) |
| 79 | \( 1 + (19.2 - 64.1i)T + (-5.20e3 - 3.44e3i)T^{2} \) |
| 83 | \( 1 + (-5.65 + 0.622i)T + (6.72e3 - 1.49e3i)T^{2} \) |
| 89 | \( 1 + (-127. + 23.5i)T + (7.39e3 - 2.82e3i)T^{2} \) |
| 97 | \( 1 + (-50.0 + 15.0i)T + (7.84e3 - 5.19e3i)T^{2} \) |
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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
−10.53839809753073260860397683665, −9.574397163915111058335994954869, −8.901273086083343443372352018500, −7.56732057791266656991309270303, −7.17973739172435972433422834320, −6.48881891905290148071100692047, −5.14071591715299977418911914067, −3.89989794271644133914870236308, −2.24253153020675986458394053562, −0.24185149160479236502784151272,
1.65586474444848105417588684054, 2.71669069897598066124082091566, 4.04448012271663864374975054519, 5.31613112521306900405238206939, 6.40157838064940685907001945193, 7.62650783676322650776697101764, 8.799490099658081561809475842668, 9.586019737470834195050696386761, 10.25849551689229628731326429497, 10.92873232383269275278457047999
