L(s) = 1 | + (1.72 + 0.143i)3-s + 2.77·5-s + (−0.855 − 2.50i)7-s + (2.95 + 0.496i)9-s + 3.43·11-s + (−0.429 + 0.743i)13-s + (4.78 + 0.398i)15-s + (−0.405 + 0.701i)17-s + (−0.750 − 1.29i)19-s + (−1.11 − 4.44i)21-s − 7.64·23-s + 2.68·25-s + (5.03 + 1.28i)27-s + (3.99 + 6.92i)29-s + (−3.60 − 6.24i)31-s + ⋯ |
L(s) = 1 | + (0.996 + 0.0829i)3-s + 1.23·5-s + (−0.323 − 0.946i)7-s + (0.986 + 0.165i)9-s + 1.03·11-s + (−0.119 + 0.206i)13-s + (1.23 + 0.102i)15-s + (−0.0982 + 0.170i)17-s + (−0.172 − 0.298i)19-s + (−0.243 − 0.969i)21-s − 1.59·23-s + 0.536·25-s + (0.969 + 0.246i)27-s + (0.742 + 1.28i)29-s + (−0.647 − 1.12i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 + 0.246i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.969 + 0.246i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.819544756\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.819544756\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.72 - 0.143i)T \) |
| 7 | \( 1 + (0.855 + 2.50i)T \) |
good | 5 | \( 1 - 2.77T + 5T^{2} \) |
| 11 | \( 1 - 3.43T + 11T^{2} \) |
| 13 | \( 1 + (0.429 - 0.743i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (0.405 - 0.701i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.750 + 1.29i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 7.64T + 23T^{2} \) |
| 29 | \( 1 + (-3.99 - 6.92i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (3.60 + 6.24i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.458 - 0.793i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.67 + 2.90i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (1.20 + 2.08i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (0.307 - 0.532i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-6.31 + 10.9i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.734 - 1.27i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (5.71 - 9.90i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-8.10 - 14.0i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 14.4T + 71T^{2} \) |
| 73 | \( 1 + (4.16 - 7.22i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (1.37 - 2.38i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-5.75 - 9.97i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (5.11 + 8.85i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.82 - 6.63i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.996123446093060037060215711143, −9.162530652969212202136395061589, −8.480261697520317817146809512698, −7.31790018669934515690095757334, −6.68250201579507789603522664379, −5.75576696715681378312169466990, −4.37075255495113931446512649109, −3.68646965191534573906574863178, −2.38290706025642774765725437928, −1.41344978148871608032415037247,
1.67237964655278979335047252589, 2.42517624906658818711334947101, 3.51355587987734014551733540254, 4.69090880023389250107082836766, 6.02367617647483052797551659382, 6.35388269839857513235843169349, 7.61662847171765251460147727471, 8.555069860654103058242759922289, 9.235813359557352498480434065887, 9.752008572500324785695421424415