| L(s) = 1 | + 3i·3-s + (5 − 10i)5-s + 4i·7-s − 9·9-s + 28·11-s − 16i·13-s + (30 + 15i)15-s − 108i·17-s + 32·19-s − 12·21-s + 28i·23-s + (−75 − 100i)25-s − 27i·27-s + 238·29-s + 180·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + (0.447 − 0.894i)5-s + 0.215i·7-s − 0.333·9-s + 0.767·11-s − 0.341i·13-s + (0.516 + 0.258i)15-s − 1.54i·17-s + 0.386·19-s − 0.124·21-s + 0.253i·23-s + (−0.599 − 0.800i)25-s − 0.192i·27-s + 1.52·29-s + 1.04·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.90022 - 0.448582i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.90022 - 0.448582i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3iT \) |
| 5 | \( 1 + (-5 + 10i)T \) |
| good | 7 | \( 1 - 4iT - 343T^{2} \) |
| 11 | \( 1 - 28T + 1.33e3T^{2} \) |
| 13 | \( 1 + 16iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 108iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 32T + 6.85e3T^{2} \) |
| 23 | \( 1 - 28iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 238T + 2.43e4T^{2} \) |
| 31 | \( 1 - 180T + 2.97e4T^{2} \) |
| 37 | \( 1 - 40iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 422T + 6.89e4T^{2} \) |
| 43 | \( 1 + 276iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 60iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 220iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 804T + 2.05e5T^{2} \) |
| 61 | \( 1 + 358T + 2.26e5T^{2} \) |
| 67 | \( 1 + 884iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 64T + 3.57e5T^{2} \) |
| 73 | \( 1 + 152iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 932T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.29e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 1.14e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 824iT - 9.12e5T^{2} \) |
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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
−11.79375861184102756700748526799, −10.52825044424822434810864330674, −9.477212736153836855739827678743, −9.005167162089953398063169195783, −7.80560201457035987215768291110, −6.35512758739217660217845337291, −5.24330595345181253775667435599, −4.36599882740992326790353609372, −2.77595394874203796839200201053, −0.916639085499223016696018984617,
1.38914783662101250070884485951, 2.80950935318641307171717475215, 4.21969795498280073977727278028, 6.00837591776011731007855101178, 6.58661833916476965887301379413, 7.66685553852563841546974788215, 8.781598239961430720847544262961, 9.958149482380564699630390111721, 10.79591764249374451171075920018, 11.76327068350993535229702077340
