L(s) = 1 | − 3.86i·5-s − 3.73i·7-s − 1.03·11-s − 4.46·13-s − 1.79i·17-s − 1.73i·19-s + 8.76·23-s − 9.92·25-s − 7.72i·29-s + 7.46i·31-s − 14.4·35-s − 0.464·37-s + 7.72i·41-s − 0.535i·43-s − 4.62·47-s + ⋯ |
L(s) = 1 | − 1.72i·5-s − 1.41i·7-s − 0.312·11-s − 1.23·13-s − 0.434i·17-s − 0.397i·19-s + 1.82·23-s − 1.98·25-s − 1.43i·29-s + 1.34i·31-s − 2.43·35-s − 0.0762·37-s + 1.20i·41-s − 0.0817i·43-s − 0.674·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.123829562\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.123829562\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 3.86iT - 5T^{2} \) |
| 7 | \( 1 + 3.73iT - 7T^{2} \) |
| 11 | \( 1 + 1.03T + 11T^{2} \) |
| 13 | \( 1 + 4.46T + 13T^{2} \) |
| 17 | \( 1 + 1.79iT - 17T^{2} \) |
| 19 | \( 1 + 1.73iT - 19T^{2} \) |
| 23 | \( 1 - 8.76T + 23T^{2} \) |
| 29 | \( 1 + 7.72iT - 29T^{2} \) |
| 31 | \( 1 - 7.46iT - 31T^{2} \) |
| 37 | \( 1 + 0.464T + 37T^{2} \) |
| 41 | \( 1 - 7.72iT - 41T^{2} \) |
| 43 | \( 1 + 0.535iT - 43T^{2} \) |
| 47 | \( 1 + 4.62T + 47T^{2} \) |
| 53 | \( 1 - 3.58iT - 53T^{2} \) |
| 59 | \( 1 - 12.3T + 59T^{2} \) |
| 61 | \( 1 + 11.3T + 61T^{2} \) |
| 67 | \( 1 - 6.26iT - 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 + 3.92T + 73T^{2} \) |
| 79 | \( 1 + 4.80iT - 79T^{2} \) |
| 83 | \( 1 + 2.07T + 83T^{2} \) |
| 89 | \( 1 + 1.79iT - 89T^{2} \) |
| 97 | \( 1 - 7T + 97T^{2} \) |
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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
−8.992816851185250505796516615242, −8.107790386131215290468131004596, −7.43331581783019381934269216113, −6.70166317063247641556203402220, −5.28711713064968465865500724804, −4.81381645751664152491280896455, −4.17875443238620745288324648036, −2.85571357245303933713055038882, −1.33379954911521026932377881978, −0.43267851944749187253922365517,
2.12999037466114420816594087130, 2.74227088568524671898105096105, 3.53886239796710554584173185250, 5.01492322305404790174305283351, 5.69754412387768907974456566432, 6.63710240532136143640847571721, 7.20634017190526409718374588541, 8.040910903355964562414068019973, 9.041785492121793512790174469912, 9.710346908469015500652115379653