L(s) = 1 | + 0.185i·3-s − 4.45i·7-s + 2.96·9-s + 2.64·11-s − 1.30i·13-s − 3.51i·17-s + 19-s + 0.826·21-s + 6.52i·23-s + 1.10i·27-s + 5.20·29-s + 10.8·31-s + 0.490i·33-s + 2.04i·37-s + 0.241·39-s + ⋯ |
L(s) = 1 | + 0.107i·3-s − 1.68i·7-s + 0.988·9-s + 0.797·11-s − 0.360i·13-s − 0.853i·17-s + 0.229·19-s + 0.180·21-s + 1.36i·23-s + 0.212i·27-s + 0.967·29-s + 1.94·31-s + 0.0854i·33-s + 0.336i·37-s + 0.0386·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.265602260\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.265602260\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 - 0.185iT - 3T^{2} \) |
| 7 | \( 1 + 4.45iT - 7T^{2} \) |
| 11 | \( 1 - 2.64T + 11T^{2} \) |
| 13 | \( 1 + 1.30iT - 13T^{2} \) |
| 17 | \( 1 + 3.51iT - 17T^{2} \) |
| 23 | \( 1 - 6.52iT - 23T^{2} \) |
| 29 | \( 1 - 5.20T + 29T^{2} \) |
| 31 | \( 1 - 10.8T + 31T^{2} \) |
| 37 | \( 1 - 2.04iT - 37T^{2} \) |
| 41 | \( 1 + 3.80T + 41T^{2} \) |
| 43 | \( 1 + 4.77iT - 43T^{2} \) |
| 47 | \( 1 - 1.49iT - 47T^{2} \) |
| 53 | \( 1 + 0.225iT - 53T^{2} \) |
| 59 | \( 1 - 2.86T + 59T^{2} \) |
| 61 | \( 1 + 6.31T + 61T^{2} \) |
| 67 | \( 1 + 13.1iT - 67T^{2} \) |
| 71 | \( 1 - 12.3T + 71T^{2} \) |
| 73 | \( 1 - 5.42iT - 73T^{2} \) |
| 79 | \( 1 + 14.9T + 79T^{2} \) |
| 83 | \( 1 + 3.84iT - 83T^{2} \) |
| 89 | \( 1 + 1.67T + 89T^{2} \) |
| 97 | \( 1 - 9.48iT - 97T^{2} \) |
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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
−8.181892987502643598263382149842, −7.55727742498734139752095880571, −6.91226890537512336053500537865, −6.44650205928610849453902103731, −5.15652668551637110641534906989, −4.46222295004069956802293413275, −3.83339299454468061361415914394, −3.02402740951813655046822517130, −1.48046136886080260873656137154, −0.790436429241349998637711532800,
1.20402078652065491988634112902, 2.17512797071609391265995140089, 2.97963990119312507033193773447, 4.19396331986799826934397194362, 4.74396943909373835547124423574, 5.79660741490129842053923833808, 6.44551850288340189648181054241, 6.91809813671018492459607508043, 8.254976954285041557007936632237, 8.444453861376146926656748844189