L(s) = 1 | − 0.584i·2-s + 3-s + 1.65·4-s − 1.95i·5-s − 0.584i·6-s − 1.51i·7-s − 2.13i·8-s + 9-s − 1.14·10-s − i·11-s + 1.65·12-s + (−3.49 + 0.887i)13-s − 0.883·14-s − 1.95i·15-s + 2.06·16-s − 3.44·17-s + ⋯ |
L(s) = 1 | − 0.413i·2-s + 0.577·3-s + 0.829·4-s − 0.873i·5-s − 0.238i·6-s − 0.571i·7-s − 0.756i·8-s + 0.333·9-s − 0.361·10-s − 0.301i·11-s + 0.478·12-s + (−0.969 + 0.246i)13-s − 0.236·14-s − 0.504i·15-s + 0.516·16-s − 0.834·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.246 + 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.246 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.54348 - 1.20057i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.54348 - 1.20057i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 11 | \( 1 + iT \) |
| 13 | \( 1 + (3.49 - 0.887i)T \) |
good | 2 | \( 1 + 0.584iT - 2T^{2} \) |
| 5 | \( 1 + 1.95iT - 5T^{2} \) |
| 7 | \( 1 + 1.51iT - 7T^{2} \) |
| 17 | \( 1 + 3.44T + 17T^{2} \) |
| 19 | \( 1 - 5.09iT - 19T^{2} \) |
| 23 | \( 1 - 0.701T + 23T^{2} \) |
| 29 | \( 1 - 5.04T + 29T^{2} \) |
| 31 | \( 1 - 7.26iT - 31T^{2} \) |
| 37 | \( 1 - 2.08iT - 37T^{2} \) |
| 41 | \( 1 - 1.03iT - 41T^{2} \) |
| 43 | \( 1 - 5.48T + 43T^{2} \) |
| 47 | \( 1 + 3.75iT - 47T^{2} \) |
| 53 | \( 1 + 0.502T + 53T^{2} \) |
| 59 | \( 1 - 2.65iT - 59T^{2} \) |
| 61 | \( 1 + 5.38T + 61T^{2} \) |
| 67 | \( 1 + 8.47iT - 67T^{2} \) |
| 71 | \( 1 + 2.31iT - 71T^{2} \) |
| 73 | \( 1 + 2.21iT - 73T^{2} \) |
| 79 | \( 1 + 5.54T + 79T^{2} \) |
| 83 | \( 1 - 14.5iT - 83T^{2} \) |
| 89 | \( 1 - 1.80iT - 89T^{2} \) |
| 97 | \( 1 - 14.5iT - 97T^{2} \) |
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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
−10.85037646187942047489180082175, −10.18491770414577745398362562435, −9.213018668749708902588414532791, −8.269059945831322955959436217463, −7.32477649566249607091491712694, −6.43666889343930551478988413221, −4.97995114603887845004452581370, −3.87817015711087439486930593427, −2.62580869438595163665181359938, −1.30745626498582484053715845528,
2.34953874392678266094555662051, 2.84064756745205897963106416112, 4.59388671215904527114519050901, 5.89730999452293035802371657272, 6.92596601527640565959811202953, 7.40281376099407010356833071325, 8.513894325224058505171232152240, 9.497065591949274685774131950980, 10.53049200547432708271435077679, 11.26710015653964920190889099012