| L(s) = 1 | + 3·3-s − 8.63i·5-s + (−18.2 + 3.01i)7-s + 9·9-s − 4.41i·11-s + 47.2i·13-s − 25.9i·15-s − 130. i·17-s − 44.8·19-s + (−54.8 + 9.03i)21-s + 104. i·23-s + 50.3·25-s + 27·27-s − 168.·29-s − 27.5·31-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.772i·5-s + (−0.986 + 0.162i)7-s + 0.333·9-s − 0.120i·11-s + 1.00i·13-s − 0.446i·15-s − 1.86i·17-s − 0.541·19-s + (−0.569 + 0.0939i)21-s + 0.948i·23-s + 0.403·25-s + 0.192·27-s − 1.07·29-s − 0.159·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.162 - 0.986i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.162 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9970598138\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9970598138\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 7 | \( 1 + (18.2 - 3.01i)T \) |
| good | 5 | \( 1 + 8.63iT - 125T^{2} \) |
| 11 | \( 1 + 4.41iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 47.2iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 130. iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 44.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 104. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 168.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 27.5T + 2.97e4T^{2} \) |
| 37 | \( 1 - 425.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 332. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 218. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 390.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 736.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 469.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 21.9iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 365. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 620. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 938. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 883. iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 739.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 66.4iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 353. iT - 9.12e5T^{2} \) |
| 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
−9.479211804280334427351389597357, −8.887246386691378350889820313775, −7.88164284201285464650343665012, −7.05851775294195269887636999191, −6.26451382255776856921989156518, −5.16279143352574776076987317153, −4.33933017961013140794666444109, −3.30877579393529116278625987537, −2.38878975538993801722082180475, −1.09838542392091296373182271221,
0.21845490901922148470105156071, 1.84780388860250719956113074414, 2.96659968697245147043276649361, 3.55097567761593707621662955539, 4.57083535534424210327647962950, 6.11421551268259908789868106572, 6.35792638541600290932085456814, 7.51857953251419580983090723415, 8.110600888331041790278111635640, 9.066431901373223660308787941122
