| L(s) = 1 | − 3-s + 3.46i·5-s + 9-s − 3.46i·11-s − 1.73i·13-s − 3.46i·15-s + 5·19-s − 6.92i·23-s − 6.99·25-s − 27-s − 5·31-s + 3.46i·33-s − 11·37-s + 1.73i·39-s − 3.46i·41-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.54i·5-s + 0.333·9-s − 1.04i·11-s − 0.480i·13-s − 0.894i·15-s + 1.14·19-s − 1.44i·23-s − 1.39·25-s − 0.192·27-s − 0.898·31-s + 0.603i·33-s − 1.80·37-s + 0.277i·39-s − 0.541i·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 + 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.147579183\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.147579183\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 3.46iT - 5T^{2} \) |
| 11 | \( 1 + 3.46iT - 11T^{2} \) |
| 13 | \( 1 + 1.73iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 5T + 19T^{2} \) |
| 23 | \( 1 + 6.92iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 5T + 31T^{2} \) |
| 37 | \( 1 + 11T + 37T^{2} \) |
| 41 | \( 1 + 3.46iT - 41T^{2} \) |
| 43 | \( 1 + 8.66iT - 43T^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 - 12T + 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 + 13.8iT - 61T^{2} \) |
| 67 | \( 1 - 8.66iT - 67T^{2} \) |
| 71 | \( 1 + 3.46iT - 71T^{2} \) |
| 73 | \( 1 - 5.19iT - 73T^{2} \) |
| 79 | \( 1 + 12.1iT - 79T^{2} \) |
| 83 | \( 1 - 18T + 83T^{2} \) |
| 89 | \( 1 - 6.92iT - 89T^{2} \) |
| 97 | \( 1 - 6.92iT - 97T^{2} \) |
| 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
−8.856140010917564065263200027539, −8.045962385903677579649840497420, −7.04840591138849773268916068226, −6.74612151642157778583310294098, −5.75565775758836832693955414795, −5.21499812717345520322333342865, −3.76191126042707419299838809967, −3.20838385293675385367088968887, −2.16570345940955167125357545982, −0.48343160271726237780202282251,
1.10397140458889933376716455222, 1.89013643503359349959142638018, 3.56493552501882482018495894551, 4.46243560852116920068933041513, 5.19485696281069393294831749671, 5.60324884093449673494405442307, 6.85976078204587706182350583504, 7.48424961501383884114538853338, 8.341952303918791598209048671146, 9.276721696535111346409947684709
