| L(s) = 1 | + (4.59 + 2.42i)3-s − 2.41i·5-s + 7i·7-s + (15.2 + 22.2i)9-s + 37.7·11-s − 24.8·13-s + (5.83 − 11.0i)15-s + 22.2i·17-s + 68.4i·19-s + (−16.9 + 32.1i)21-s + 44.4·23-s + 119.·25-s + (16.2 + 139. i)27-s + 178. i·29-s − 109. i·31-s + ⋯ |
| L(s) = 1 | + (0.884 + 0.466i)3-s − 0.215i·5-s + 0.377i·7-s + (0.565 + 0.824i)9-s + 1.03·11-s − 0.529·13-s + (0.100 − 0.190i)15-s + 0.317i·17-s + 0.826i·19-s + (−0.176 + 0.334i)21-s + 0.403·23-s + 0.953·25-s + (0.116 + 0.993i)27-s + 1.14i·29-s − 0.636i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.466 - 0.884i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.466 - 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.626216230\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.626216230\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-4.59 - 2.42i)T \) |
| 7 | \( 1 - 7iT \) |
| good | 5 | \( 1 + 2.41iT - 125T^{2} \) |
| 11 | \( 1 - 37.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 24.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 22.2iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 68.4iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 44.4T + 1.21e4T^{2} \) |
| 29 | \( 1 - 178. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 109. iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 168.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 383. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 371. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 323.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 401. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 34.0T + 2.05e5T^{2} \) |
| 61 | \( 1 - 25.4T + 2.26e5T^{2} \) |
| 67 | \( 1 - 118. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 106.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.01e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 649. iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 250.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 4.02iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 454.T + 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.21262955430959619496009339639, −10.16277346956762314809764642863, −9.311333854816332717378886009534, −8.649819971417102236790524203028, −7.66978320113491969639642014928, −6.51842909171855911651886790999, −5.14203177216306998374926482682, −4.08963852445967881276114490343, −2.96067673348780749747650606310, −1.54788332127628642474475857619,
0.928143942797596456591230828070, 2.42323522439314134773580588061, 3.58047216658148943025558113363, 4.75792530847525439663539780339, 6.48765521726891484709348189579, 7.09064701310012172852937120839, 8.111523124292740151317809151105, 9.135090572068879977227139776387, 9.774638162384240333848103853380, 11.02149945710931764153379957196
