L(s) = 1 | + (0.707 − 1.22i)2-s + (−0.999 − 1.73i)4-s + 3.46i·7-s − 2.82·8-s + 3.16·11-s − 4.47·13-s + (4.24 + 2.44i)14-s + (−2.00 + 3.46i)16-s − 6.32·17-s + (−2 + 3.87i)19-s + (2.23 − 3.87i)22-s + 5.47i·23-s + 5·25-s + (−3.16 + 5.47i)26-s + (5.99 − 3.46i)28-s + 1.41·29-s + ⋯ |
L(s) = 1 | + (0.499 − 0.866i)2-s + (−0.499 − 0.866i)4-s + 1.30i·7-s − 0.999·8-s + 0.953·11-s − 1.24·13-s + (1.13 + 0.654i)14-s + (−0.500 + 0.866i)16-s − 1.53·17-s + (−0.458 + 0.888i)19-s + (0.476 − 0.825i)22-s + 1.14i·23-s + 25-s + (−0.620 + 1.07i)26-s + (1.13 − 0.654i)28-s + 0.262·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.458 - 0.888i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.458 - 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.028656909\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.028656909\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 1.22i)T \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (2 - 3.87i)T \) |
good | 5 | \( 1 - 5T^{2} \) |
| 7 | \( 1 - 3.46iT - 7T^{2} \) |
| 11 | \( 1 - 3.16T + 11T^{2} \) |
| 13 | \( 1 + 4.47T + 13T^{2} \) |
| 17 | \( 1 + 6.32T + 17T^{2} \) |
| 23 | \( 1 - 5.47iT - 23T^{2} \) |
| 29 | \( 1 - 1.41T + 29T^{2} \) |
| 31 | \( 1 + 8.94T + 31T^{2} \) |
| 37 | \( 1 + 4.47T + 37T^{2} \) |
| 41 | \( 1 - 2.44iT - 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 5.47iT - 47T^{2} \) |
| 53 | \( 1 - 4.24T + 53T^{2} \) |
| 59 | \( 1 + 4.89iT - 59T^{2} \) |
| 61 | \( 1 - 10.3iT - 61T^{2} \) |
| 67 | \( 1 + 7.74iT - 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 + 12T + 73T^{2} \) |
| 79 | \( 1 + 4.47T + 79T^{2} \) |
| 83 | \( 1 - 15.8T + 83T^{2} \) |
| 89 | \( 1 - 17.1iT - 89T^{2} \) |
| 97 | \( 1 + 15.4iT - 97T^{2} \) |
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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
−9.612549164849985230605535607009, −9.148709622855554642300584860684, −8.508990198784454971286445740911, −7.11475622131473451257136178790, −6.21952216041146914158712312692, −5.40293137256400105181750905176, −4.61435453581979584980780912071, −3.59906498643372312398371076186, −2.49502210378172303780187543777, −1.72146450280527091679784151152,
0.33605209465174742598442123407, 2.37578242542297392651074979599, 3.71483568082702740303754892285, 4.45620135782952356698681839059, 5.05524401416802407371757564597, 6.51054670218153364787813263388, 6.90408974546402786167154570600, 7.46511639437992713802876708388, 8.723866923983568627481418772116, 9.084684327136900152159594613337