L(s) = 1 | + (1.85 − 1.14i)2-s + (0.597 + 2.10i)3-s + (1.23 − 2.48i)4-s + (−3.45 + 1.33i)5-s + (3.52 + 3.21i)6-s + (0.481 − 5.19i)7-s + (−0.155 − 1.67i)8-s + (−1.50 + 0.933i)9-s + (−4.87 + 6.45i)10-s + (−1.36 + 0.846i)11-s + (5.95 + 1.11i)12-s + (0.000230 − 0.00248i)13-s + (−5.07 − 10.1i)14-s + (−4.87 − 6.45i)15-s + (1.12 + 1.48i)16-s + (−2.22 + 2.02i)17-s + ⋯ |
L(s) = 1 | + (1.31 − 0.813i)2-s + (0.345 + 1.21i)3-s + (0.617 − 1.24i)4-s + (−1.54 + 0.598i)5-s + (1.43 + 1.31i)6-s + (0.181 − 1.96i)7-s + (−0.0549 − 0.592i)8-s + (−0.502 + 0.311i)9-s + (−1.54 + 2.04i)10-s + (−0.412 + 0.255i)11-s + (1.71 + 0.321i)12-s + (6.38e−5 − 0.000689i)13-s + (−1.35 − 2.72i)14-s + (−1.25 − 1.66i)15-s + (0.280 + 0.371i)16-s + (−0.539 + 0.491i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.69285 - 0.196645i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.69285 - 0.196645i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 103 | \( 1 + (-5.59 - 8.46i)T \) |
good | 2 | \( 1 + (-1.85 + 1.14i)T + (0.891 - 1.79i)T^{2} \) |
| 3 | \( 1 + (-0.597 - 2.10i)T + (-2.55 + 1.57i)T^{2} \) |
| 5 | \( 1 + (3.45 - 1.33i)T + (3.69 - 3.36i)T^{2} \) |
| 7 | \( 1 + (-0.481 + 5.19i)T + (-6.88 - 1.28i)T^{2} \) |
| 11 | \( 1 + (1.36 - 0.846i)T + (4.90 - 9.84i)T^{2} \) |
| 13 | \( 1 + (-0.000230 + 0.00248i)T + (-12.7 - 2.38i)T^{2} \) |
| 17 | \( 1 + (2.22 - 2.02i)T + (1.56 - 16.9i)T^{2} \) |
| 19 | \( 1 + (0.292 - 1.02i)T + (-16.1 - 10.0i)T^{2} \) |
| 23 | \( 1 + (1.41 + 0.873i)T + (10.2 + 20.5i)T^{2} \) |
| 29 | \( 1 + (-3.63 + 1.40i)T + (21.4 - 19.5i)T^{2} \) |
| 31 | \( 1 + (-2.29 - 3.04i)T + (-8.48 + 29.8i)T^{2} \) |
| 37 | \( 1 + (9.36 + 1.75i)T + (34.5 + 13.3i)T^{2} \) |
| 41 | \( 1 + (-3.20 - 1.23i)T + (30.2 + 27.6i)T^{2} \) |
| 43 | \( 1 + (-5.93 + 1.10i)T + (40.0 - 15.5i)T^{2} \) |
| 47 | \( 1 + 0.110T + 47T^{2} \) |
| 53 | \( 1 + (-1.88 + 6.63i)T + (-45.0 - 27.9i)T^{2} \) |
| 59 | \( 1 + (0.0110 + 0.119i)T + (-57.9 + 10.8i)T^{2} \) |
| 61 | \( 1 + (-3.95 + 3.60i)T + (5.62 - 60.7i)T^{2} \) |
| 67 | \( 1 + (-0.127 - 1.37i)T + (-65.8 + 12.3i)T^{2} \) |
| 71 | \( 1 + (11.5 + 4.47i)T + (52.4 + 47.8i)T^{2} \) |
| 73 | \( 1 + (3.45 + 1.33i)T + (53.9 + 49.1i)T^{2} \) |
| 79 | \( 1 + (2.59 - 1.00i)T + (58.3 - 53.2i)T^{2} \) |
| 83 | \( 1 + (0.428 - 4.62i)T + (-81.5 - 15.2i)T^{2} \) |
| 89 | \( 1 + (2.68 + 5.39i)T + (-53.6 + 71.0i)T^{2} \) |
| 97 | \( 1 + (10.1 + 9.28i)T + (8.95 + 96.5i)T^{2} \) |
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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
−13.96578530812703290581166781946, −12.74237377773905569897735753003, −11.54376539034294673687329219212, −10.58751847086753357468289435017, −10.36587207049692027742536863914, −8.194953291805224489145432165546, −6.97798115273155490781957173594, −4.62727035975905419031649466193, −4.06951325830589350017368813222, −3.33332072273614712408720472823,
2.81331359250055138581742365897, 4.59360356081688743630005507971, 5.75976993957842636667484613423, 7.06252334433366368839691541706, 8.046524273127581660384387644613, 8.783323566388634389378972507378, 11.60482327667672866614675953164, 12.22526222010552878198906493770, 12.74524863927569758046809245456, 13.77891086462304678274262848169