| L(s) = 1 | − i·3-s + (1.41 − 1.73i)5-s − 9-s + 3.46i·13-s + (−1.73 − 1.41i)15-s + 4.89i·17-s + 4.89·19-s − 2.82i·23-s + (−0.999 − 4.89i)25-s + i·27-s + 2.82·29-s + 6.92·31-s + 3.46i·37-s + 3.46·39-s − 6·41-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + (0.632 − 0.774i)5-s − 0.333·9-s + 0.960i·13-s + (−0.447 − 0.365i)15-s + 1.18i·17-s + 1.12·19-s − 0.589i·23-s + (−0.199 − 0.979i)25-s + 0.192i·27-s + 0.525·29-s + 1.24·31-s + 0.569i·37-s + 0.554·39-s − 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.774 + 0.632i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.774 + 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.237986744\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.237986744\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + (-1.41 + 1.73i)T \) |
| good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 3.46iT - 13T^{2} \) |
| 17 | \( 1 - 4.89iT - 17T^{2} \) |
| 19 | \( 1 - 4.89T + 19T^{2} \) |
| 23 | \( 1 + 2.82iT - 23T^{2} \) |
| 29 | \( 1 - 2.82T + 29T^{2} \) |
| 31 | \( 1 - 6.92T + 31T^{2} \) |
| 37 | \( 1 - 3.46iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + 2.82iT - 47T^{2} \) |
| 53 | \( 1 - 3.46iT - 53T^{2} \) |
| 59 | \( 1 - 9.79T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 - 13.8T + 71T^{2} \) |
| 73 | \( 1 + 9.79iT - 73T^{2} \) |
| 79 | \( 1 - 6.92T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 9.79iT - 97T^{2} \) |
| 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
−8.416872373118613403467220639032, −7.82328674390152950503995212986, −6.70851859035070992057056174133, −6.35087394799734765583878707365, −5.43445595511319270778664980909, −4.72820910613869399675968917979, −3.84198278616864295156472062736, −2.64981538434759426551738271416, −1.74696708506659694856654671988, −0.914764438254787483561646218315,
0.879239655229458385454513397398, 2.38590857604412673994413893178, 3.04768168001008466284469089244, 3.78722168938126828761072258300, 5.07850049055178694190401607301, 5.39703579661239322179081961515, 6.31750248478134919141201589470, 7.12398008811492871628729553094, 7.74222480320770565287993015648, 8.677565310832719125770129245234
