L(s) = 1 | + (−0.258 − 0.965i)3-s + (−1.73 − i)4-s + (0.189 + 2.63i)7-s + (−0.866 + 0.499i)9-s + (−3 + 5.19i)11-s + (−0.517 + 1.93i)12-s + (1.41 − 1.41i)13-s + (1.99 + 3.46i)16-s + (5.79 − 1.55i)17-s + (1.73 + 3i)19-s + (2.50 − 0.866i)21-s + (0.707 + 0.707i)27-s + (2.31 − 4.76i)28-s + (−4.5 − 2.59i)31-s + (5.79 + 1.55i)33-s + ⋯ |
L(s) = 1 | + (−0.149 − 0.557i)3-s + (−0.866 − 0.5i)4-s + (0.0716 + 0.997i)7-s + (−0.288 + 0.166i)9-s + (−0.904 + 1.56i)11-s + (−0.149 + 0.557i)12-s + (0.392 − 0.392i)13-s + (0.499 + 0.866i)16-s + (1.40 − 0.376i)17-s + (0.397 + 0.688i)19-s + (0.545 − 0.188i)21-s + (0.136 + 0.136i)27-s + (0.436 − 0.899i)28-s + (−0.808 − 0.466i)31-s + (1.00 + 0.270i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.657 - 0.753i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.657 - 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.837652 + 0.380707i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.837652 + 0.380707i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.258 + 0.965i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.189 - 2.63i)T \) |
good | 2 | \( 1 + (1.73 + i)T^{2} \) |
| 11 | \( 1 + (3 - 5.19i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.41 + 1.41i)T - 13iT^{2} \) |
| 17 | \( 1 + (-5.79 + 1.55i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-1.73 - 3i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + (4.5 + 2.59i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-8.36 - 2.24i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 - 10.3iT - 41T^{2} \) |
| 43 | \( 1 + (-3.67 - 3.67i)T + 43iT^{2} \) |
| 47 | \( 1 + (1.55 - 5.79i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (10.0 - 2.68i)T + (45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (5.19 - 9i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-7.5 + 4.33i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.896 + 3.34i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + (-0.258 - 0.965i)T + (-63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-4.33 + 2.5i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (4.24 - 4.24i)T - 83iT^{2} \) |
| 89 | \( 1 + (-5.19 - 9i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (9.19 + 9.19i)T + 97iT^{2} \) |
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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
−10.94619590327040353908772959928, −9.803415699825598353392050240605, −9.488429414726341352568459379817, −8.085582044470746979643493311694, −7.67336427236289678577036459303, −6.12060755833589788265834748330, −5.43919187201726366447868786749, −4.58174004981262061270803164047, −2.91367530668217284890669431162, −1.46398499375201067928077764017,
0.61901554323764199639669016041, 3.22816677713087200743535681577, 3.84656270687051038701794293506, 5.03765289579696803967019525471, 5.84566665354532054381244768689, 7.35172259218114549379334635876, 8.150932355695101477601578565263, 8.940658782291273901288401336372, 9.894394062281860929178178832816, 10.72119035529936763936322379444