L(s) = 1 | + (2.41 − 0.708i)2-s + (−1.04 − 1.20i)3-s + (3.63 − 2.33i)4-s + (−0.210 − 1.46i)5-s + (−3.37 − 2.16i)6-s + (−0.510 + 1.11i)7-s + (3.81 − 4.40i)8-s + (0.0651 − 0.453i)9-s + (−1.54 − 3.38i)10-s + (−3.40 − 0.999i)11-s + (−6.60 − 1.93i)12-s + (−0.0566 − 0.123i)13-s + (−0.439 + 3.05i)14-s + (−1.54 + 1.78i)15-s + (2.49 − 5.45i)16-s + (5.07 + 3.26i)17-s + ⋯ |
L(s) = 1 | + (1.70 − 0.500i)2-s + (−0.602 − 0.695i)3-s + (1.81 − 1.16i)4-s + (−0.0941 − 0.654i)5-s + (−1.37 − 0.884i)6-s + (−0.192 + 0.422i)7-s + (1.34 − 1.55i)8-s + (0.0217 − 0.151i)9-s + (−0.488 − 1.06i)10-s + (−1.02 − 0.301i)11-s + (−1.90 − 0.559i)12-s + (−0.0157 − 0.0343i)13-s + (−0.117 + 0.816i)14-s + (−0.398 + 0.460i)15-s + (0.623 − 1.36i)16-s + (1.23 + 0.791i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.415 + 0.909i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.415 + 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.59313 - 2.48012i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.59313 - 2.48012i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 23 | \( 1 \) |
good | 2 | \( 1 + (-2.41 + 0.708i)T + (1.68 - 1.08i)T^{2} \) |
| 3 | \( 1 + (1.04 + 1.20i)T + (-0.426 + 2.96i)T^{2} \) |
| 5 | \( 1 + (0.210 + 1.46i)T + (-4.79 + 1.40i)T^{2} \) |
| 7 | \( 1 + (0.510 - 1.11i)T + (-4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (3.40 + 0.999i)T + (9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (0.0566 + 0.123i)T + (-8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (-5.07 - 3.26i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (-2.70 + 1.74i)T + (7.89 - 17.2i)T^{2} \) |
| 29 | \( 1 + (-3.86 - 2.48i)T + (12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (1.88 - 2.17i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (-0.554 + 3.85i)T + (-35.5 - 10.4i)T^{2} \) |
| 41 | \( 1 + (0.145 + 1.01i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (2.07 + 2.39i)T + (-6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 - 0.00935T + 47T^{2} \) |
| 53 | \( 1 + (-3.90 + 8.55i)T + (-34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (-1.93 - 4.24i)T + (-38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (8.87 - 10.2i)T + (-8.68 - 60.3i)T^{2} \) |
| 67 | \( 1 + (-9.33 + 2.74i)T + (56.3 - 36.2i)T^{2} \) |
| 71 | \( 1 + (-0.668 + 0.196i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (4.46 - 2.86i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (-4.45 - 9.76i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (0.249 - 1.73i)T + (-79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (5.76 + 6.65i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (-2.56 - 17.8i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−10.97309066409183987946966573446, −10.13271012576475264382834041171, −8.742191065096200512031632594021, −7.49682632968288666544633937880, −6.47705215962963259347284116086, −5.54996247170632643950152558650, −5.16207924640185074400106752440, −3.77771769731580653641414544371, −2.71506516095939586296332333117, −1.19508596579069580197758046367,
2.70287455757809827391491726148, 3.60044813456785580776990054560, 4.74839081186234671094817254646, 5.30476502736160220163992074953, 6.22293281319718604629425058657, 7.30483917562953291540798885039, 7.83123255756943278060270049625, 9.819049103839840963043677727037, 10.46977380972878015390970731511, 11.32886113090527487971291344362