L(s) = 1 | + (1.95 + 0.136i)2-s + (−0.0107 − 0.308i)3-s + (1.83 + 0.258i)4-s + (−0.891 + 2.05i)5-s + (0.0211 − 0.606i)6-s + (1.49 + 2.59i)7-s + (−0.275 − 0.0585i)8-s + (2.89 − 0.202i)9-s + (−2.02 + 3.89i)10-s + (−0.0475 − 0.452i)11-s + (0.0599 − 0.570i)12-s + (0.885 + 1.31i)13-s + (2.57 + 5.28i)14-s + (0.642 + 0.253i)15-s + (−4.10 − 1.17i)16-s + (0.654 − 1.61i)17-s + ⋯ |
L(s) = 1 | + (1.38 + 0.0968i)2-s + (−0.00622 − 0.178i)3-s + (0.919 + 0.129i)4-s + (−0.398 + 0.917i)5-s + (0.00864 − 0.247i)6-s + (0.565 + 0.979i)7-s + (−0.0974 − 0.0207i)8-s + (0.965 − 0.0675i)9-s + (−0.641 + 1.23i)10-s + (−0.0143 − 0.136i)11-s + (0.0173 − 0.164i)12-s + (0.245 + 0.364i)13-s + (0.688 + 1.41i)14-s + (0.165 + 0.0653i)15-s + (−1.02 − 0.293i)16-s + (0.158 − 0.392i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.723 - 0.690i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.723 - 0.690i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.60648 + 1.04395i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.60648 + 1.04395i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (0.891 - 2.05i)T \) |
| 19 | \( 1 + (-3.63 - 2.40i)T \) |
good | 2 | \( 1 + (-1.95 - 0.136i)T + (1.98 + 0.278i)T^{2} \) |
| 3 | \( 1 + (0.0107 + 0.308i)T + (-2.99 + 0.209i)T^{2} \) |
| 7 | \( 1 + (-1.49 - 2.59i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (0.0475 + 0.452i)T + (-10.7 + 2.28i)T^{2} \) |
| 13 | \( 1 + (-0.885 - 1.31i)T + (-4.86 + 12.0i)T^{2} \) |
| 17 | \( 1 + (-0.654 + 1.61i)T + (-12.2 - 11.8i)T^{2} \) |
| 23 | \( 1 + (2.34 + 2.26i)T + (0.802 + 22.9i)T^{2} \) |
| 29 | \( 1 + (-0.323 - 0.801i)T + (-20.8 + 20.1i)T^{2} \) |
| 31 | \( 1 + (-2.41 + 2.68i)T + (-3.24 - 30.8i)T^{2} \) |
| 37 | \( 1 + (1.72 + 1.25i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-1.75 - 0.503i)T + (34.7 + 21.7i)T^{2} \) |
| 43 | \( 1 + (-0.372 + 2.11i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (3.93 + 9.74i)T + (-33.8 + 32.6i)T^{2} \) |
| 53 | \( 1 + (1.63 + 0.229i)T + (50.9 + 14.6i)T^{2} \) |
| 59 | \( 1 + (0.240 + 0.965i)T + (-52.0 + 27.6i)T^{2} \) |
| 61 | \( 1 + (3.78 + 3.65i)T + (2.12 + 60.9i)T^{2} \) |
| 67 | \( 1 + (-2.72 - 1.70i)T + (29.3 + 60.2i)T^{2} \) |
| 71 | \( 1 + (12.4 + 6.59i)T + (39.7 + 58.8i)T^{2} \) |
| 73 | \( 1 + (0.376 - 0.557i)T + (-27.3 - 67.6i)T^{2} \) |
| 79 | \( 1 + (-0.311 - 8.93i)T + (-78.8 + 5.51i)T^{2} \) |
| 83 | \( 1 + (7.87 - 8.75i)T + (-8.67 - 82.5i)T^{2} \) |
| 89 | \( 1 + (-7.72 + 2.21i)T + (75.4 - 47.1i)T^{2} \) |
| 97 | \( 1 + (-3.15 + 1.97i)T + (42.5 - 87.1i)T^{2} \) |
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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
−11.70213031609685640480460849255, −10.45486418913335940021550210247, −9.399042103142779409541430402401, −8.164573836480328688784033008424, −7.14688007927042100177088415300, −6.31263725330726012481662610402, −5.39051049506487401821828379478, −4.32974335509622470209731638248, −3.36541720549460437420952826804, −2.17433440333535150511755721910,
1.36584111635707941128559978947, 3.35391382709232852816606185888, 4.35133447536001981431292023809, 4.75813670311079052203331631729, 5.85268863979284184882426466293, 7.17844121479097846980913640005, 7.962129205868793203972660920704, 9.158764496648787021619350595984, 10.20174060663271623465767721213, 11.19038511953620380010996815907