| L(s) = 1 | + i·2-s − i·3-s − 4-s − 1.78i·5-s + 6-s − i·8-s − 9-s + 1.78·10-s + (−3.15 + 1.02i)11-s + i·12-s − 6.37·13-s − 1.78·15-s + 16-s + 0.106·17-s − i·18-s − 4.15·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.577i·3-s − 0.5·4-s − 0.799i·5-s + 0.408·6-s − 0.353i·8-s − 0.333·9-s + 0.565·10-s + (−0.951 + 0.308i)11-s + 0.288i·12-s − 1.76·13-s − 0.461·15-s + 0.250·16-s + 0.0257·17-s − 0.235i·18-s − 0.952·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.517 - 0.855i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.517 - 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.041908339\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.041908339\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + iT \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (3.15 - 1.02i)T \) |
| good | 5 | \( 1 + 1.78iT - 5T^{2} \) |
| 13 | \( 1 + 6.37T + 13T^{2} \) |
| 17 | \( 1 - 0.106T + 17T^{2} \) |
| 19 | \( 1 + 4.15T + 19T^{2} \) |
| 23 | \( 1 - 7.95T + 23T^{2} \) |
| 29 | \( 1 - 7.65iT - 29T^{2} \) |
| 31 | \( 1 - 1.27iT - 31T^{2} \) |
| 37 | \( 1 - 4.52T + 37T^{2} \) |
| 41 | \( 1 + 0.0321T + 41T^{2} \) |
| 43 | \( 1 - 6.87iT - 43T^{2} \) |
| 47 | \( 1 + 9.88iT - 47T^{2} \) |
| 53 | \( 1 + 0.627T + 53T^{2} \) |
| 59 | \( 1 - 12.4iT - 59T^{2} \) |
| 61 | \( 1 - 9.95T + 61T^{2} \) |
| 67 | \( 1 - 10.8T + 67T^{2} \) |
| 71 | \( 1 + 8.42T + 71T^{2} \) |
| 73 | \( 1 - 0.125T + 73T^{2} \) |
| 79 | \( 1 + 10.1iT - 79T^{2} \) |
| 83 | \( 1 - 12.0T + 83T^{2} \) |
| 89 | \( 1 - 5.36iT - 89T^{2} \) |
| 97 | \( 1 + 15.7iT - 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.676760947056153605212366546658, −7.952425000304627681796349478048, −7.17585188352277240123953555601, −6.83898593770415698694931376330, −5.64403536942320090401466202178, −4.95603780501217510152813779395, −4.61604483800886964070840030329, −3.10052369019429385915913024920, −2.18396153745287711685133010665, −0.822788688104486912814996981504,
0.42461310672189905889389829671, 2.40282634825608693951127441019, 2.65838835476624276916017476591, 3.67629210289720820137994137650, 4.70336954610067507766406771311, 5.16326242389582511036384780868, 6.21958328018690573866282510694, 7.13402031071334460756084087566, 7.85232367146115284086125249459, 8.653798481696369945000038146921
