L(s) = 1 | + (0.0861 − 1.41i)2-s + (−1.98 − 0.243i)4-s + (0.707 − 0.707i)5-s + 2.76i·7-s + (−0.514 + 2.78i)8-s + (−0.937 − 1.05i)10-s + (3.51 − 3.51i)11-s + (−4.55 − 4.55i)13-s + (3.90 + 0.238i)14-s + (3.88 + 0.965i)16-s + 5.00·17-s + (−0.812 − 0.812i)19-s + (−1.57 + 1.23i)20-s + (−4.65 − 5.25i)22-s − 7.48i·23-s + ⋯ |
L(s) = 1 | + (0.0609 − 0.998i)2-s + (−0.992 − 0.121i)4-s + (0.316 − 0.316i)5-s + 1.04i·7-s + (−0.181 + 0.983i)8-s + (−0.296 − 0.334i)10-s + (1.05 − 1.05i)11-s + (−1.26 − 1.26i)13-s + (1.04 + 0.0636i)14-s + (0.970 + 0.241i)16-s + 1.21·17-s + (−0.186 − 0.186i)19-s + (−0.352 + 0.275i)20-s + (−0.991 − 1.12i)22-s − 1.56i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.594 + 0.804i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.594 + 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.620792 - 1.23092i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.620792 - 1.23092i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.0861 + 1.41i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.707 + 0.707i)T \) |
good | 7 | \( 1 - 2.76iT - 7T^{2} \) |
| 11 | \( 1 + (-3.51 + 3.51i)T - 11iT^{2} \) |
| 13 | \( 1 + (4.55 + 4.55i)T + 13iT^{2} \) |
| 17 | \( 1 - 5.00T + 17T^{2} \) |
| 19 | \( 1 + (0.812 + 0.812i)T + 19iT^{2} \) |
| 23 | \( 1 + 7.48iT - 23T^{2} \) |
| 29 | \( 1 + (6.03 + 6.03i)T + 29iT^{2} \) |
| 31 | \( 1 - 7.58T + 31T^{2} \) |
| 37 | \( 1 + (-1.08 + 1.08i)T - 37iT^{2} \) |
| 41 | \( 1 - 3.15iT - 41T^{2} \) |
| 43 | \( 1 + (3.10 - 3.10i)T - 43iT^{2} \) |
| 47 | \( 1 - 2.76T + 47T^{2} \) |
| 53 | \( 1 + (6.41 - 6.41i)T - 53iT^{2} \) |
| 59 | \( 1 + (-5.13 + 5.13i)T - 59iT^{2} \) |
| 61 | \( 1 + (2.49 + 2.49i)T + 61iT^{2} \) |
| 67 | \( 1 + (3.14 + 3.14i)T + 67iT^{2} \) |
| 71 | \( 1 + 3.50iT - 71T^{2} \) |
| 73 | \( 1 - 14.6iT - 73T^{2} \) |
| 79 | \( 1 - 8.95T + 79T^{2} \) |
| 83 | \( 1 + (-2.86 - 2.86i)T + 83iT^{2} \) |
| 89 | \( 1 + 7.23iT - 89T^{2} \) |
| 97 | \( 1 + 8.24T + 97T^{2} \) |
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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
−10.01027550693425677761602647982, −9.476131820952346335892897942557, −8.548419628089878195898517134337, −7.936247109177535216952129980399, −6.17437361263606799953100115398, −5.51896686406791183687577329237, −4.54576580927102264986582493240, −3.20074416797304444737617085817, −2.37999238958374438846783609540, −0.76910358289905036605869069879,
1.56079164239271494737924116737, 3.57810799088159817710183353312, 4.40801918185281712718431661721, 5.35664112151914414018351894189, 6.60349814887235927157496591268, 7.17487635215323691355101056129, 7.69866818219330274303370229430, 9.167258351895788987273663105169, 9.676090265894956020494769227238, 10.31737334878277528751319709963