L(s) = 1 | + (−1.80 + 1.31i)2-s + (1 + 3.07i)3-s + (0.927 − 2.85i)4-s + (1.61 + 1.17i)5-s + (−5.85 − 4.25i)6-s + (−0.309 + 0.951i)7-s + (0.690 + 2.12i)8-s + (−6.04 + 4.39i)9-s − 4.47·10-s + 9.70·12-s + (−1 + 0.726i)13-s + (−0.690 − 2.12i)14-s + (−2 + 6.15i)15-s + (0.809 + 0.587i)16-s + (1 + 0.726i)17-s + (5.16 − 15.8i)18-s + ⋯ |
L(s) = 1 | + (−1.27 + 0.929i)2-s + (0.577 + 1.77i)3-s + (0.463 − 1.42i)4-s + (0.723 + 0.525i)5-s + (−2.38 − 1.73i)6-s + (−0.116 + 0.359i)7-s + (0.244 + 0.751i)8-s + (−2.01 + 1.46i)9-s − 1.41·10-s + 2.80·12-s + (−0.277 + 0.201i)13-s + (−0.184 − 0.568i)14-s + (−0.516 + 1.58i)15-s + (0.202 + 0.146i)16-s + (0.242 + 0.176i)17-s + (1.21 − 3.74i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.485111 - 0.637729i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.485111 - 0.637729i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (0.309 - 0.951i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (1.80 - 1.31i)T + (0.618 - 1.90i)T^{2} \) |
| 3 | \( 1 + (-1 - 3.07i)T + (-2.42 + 1.76i)T^{2} \) |
| 5 | \( 1 + (-1.61 - 1.17i)T + (1.54 + 4.75i)T^{2} \) |
| 13 | \( 1 + (1 - 0.726i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-1 - 0.726i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-0.763 - 2.35i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + 6.47T + 23T^{2} \) |
| 29 | \( 1 + (-0.145 + 0.449i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-5.85 + 4.25i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-0.145 + 0.449i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-2.09 - 6.43i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 + (-2.23 - 6.88i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (6.85 - 4.97i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-1 + 3.07i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (2.23 + 1.62i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 - 5.52T + 67T^{2} \) |
| 71 | \( 1 + (-1.23 - 0.898i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-1.61 + 4.97i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-7.23 + 5.25i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-12.4 - 9.06i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 + (-7.61 + 5.53i)T + (29.9 - 92.2i)T^{2} \) |
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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
−10.23653214742130596922718819202, −9.661904217952648679591147180796, −9.415264311942528232791721890432, −8.279151388154642274254787080884, −7.891964678286185120076520369753, −6.35213895544602688783696046210, −5.83488804446860787736878616851, −4.65784773074121400993553468477, −3.46560500483535225992482693794, −2.24692234904594493047724476117,
0.54420762736703976741782352982, 1.57371022906192222282030887624, 2.32065376511929835661828790468, 3.34663882314299484026709262457, 5.38332104307928491092898709794, 6.50616355943612034167987669765, 7.36395141035741389167215403531, 8.117867486651597255175548256309, 8.704408052343327731738367895348, 9.514415009311939669021536001268