| L(s) = 1 | + 3-s + (2 − 1.73i)7-s + 9-s − 3.46i·11-s + 6.92i·17-s + 4·19-s + (2 − 1.73i)21-s − 3.46i·23-s + 5·25-s + 27-s − 6·29-s − 4·31-s − 3.46i·33-s − 2·37-s + 6.92i·41-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + (0.755 − 0.654i)7-s + 0.333·9-s − 1.04i·11-s + 1.68i·17-s + 0.917·19-s + (0.436 − 0.377i)21-s − 0.722i·23-s + 25-s + 0.192·27-s − 1.11·29-s − 0.718·31-s − 0.603i·33-s − 0.328·37-s + 1.08i·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 + 0.327i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.944 + 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.68282 - 0.283217i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.68282 - 0.283217i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2 + 1.73i)T \) |
| good | 5 | \( 1 - 5T^{2} \) |
| 11 | \( 1 + 3.46iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 6.92iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 3.46iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 6.92iT - 41T^{2} \) |
| 43 | \( 1 - 3.46iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 + 13.8iT - 61T^{2} \) |
| 67 | \( 1 - 3.46iT - 67T^{2} \) |
| 71 | \( 1 - 10.3iT - 71T^{2} \) |
| 73 | \( 1 - 13.8iT - 73T^{2} \) |
| 79 | \( 1 - 10.3iT - 79T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 + 6.92iT - 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
| 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
−11.21402393019801092828896068121, −10.75631764196248097395689224306, −9.612578002731000667636479215484, −8.485771049499157265068640514477, −7.957127013324096108962298835021, −6.79884022610016342374686074491, −5.56543512794058939431826308624, −4.26121407761901792491751471376, −3.20748397302295968520835610073, −1.46626636147716896102162455597,
1.83268115423352925934717375052, 3.11331228195209380581604489004, 4.65971280161067820814415392566, 5.48062793122691683606992864380, 7.17502642366272251415832094092, 7.65621549131716132561940402539, 9.058861198097951382818317190748, 9.413120518484657714453714376336, 10.69715468469419836824879919078, 11.72975023723518757745496082251
