| L(s) = 1 | + 3i·3-s + 7i·7-s − 9·9-s + 4·11-s + 54i·13-s + 14i·17-s − 92·19-s − 21·21-s − 152i·23-s − 27i·27-s + 106·29-s − 144·31-s + 12i·33-s − 158i·37-s − 162·39-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 0.377i·7-s − 0.333·9-s + 0.109·11-s + 1.15i·13-s + 0.199i·17-s − 1.11·19-s − 0.218·21-s − 1.37i·23-s − 0.192i·27-s + 0.678·29-s − 0.834·31-s + 0.0633i·33-s − 0.702i·37-s − 0.665·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8980327255\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8980327255\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - 7iT \) |
| good | 11 | \( 1 - 4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 54iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 14iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 92T + 6.85e3T^{2} \) |
| 23 | \( 1 + 152iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 106T + 2.43e4T^{2} \) |
| 31 | \( 1 + 144T + 2.97e4T^{2} \) |
| 37 | \( 1 + 158iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 390T + 6.89e4T^{2} \) |
| 43 | \( 1 + 508iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 528iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 606iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 364T + 2.05e5T^{2} \) |
| 61 | \( 1 - 678T + 2.26e5T^{2} \) |
| 67 | \( 1 + 844iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 8T + 3.57e5T^{2} \) |
| 73 | \( 1 + 422iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 384T + 4.93e5T^{2} \) |
| 83 | \( 1 + 548iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 1.19e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.50e3iT - 9.12e5T^{2} \) |
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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
−8.873710292611585553740468772235, −8.014987727968908900509961073315, −6.86202624460057260219195230945, −6.32606465557654087179360601218, −5.34907356905472767499163831716, −4.45315358010991380576863299478, −3.86524136341126907758285643072, −2.64518240234891050233234077620, −1.78937730751274217184649329827, −0.20740427915651909944462589503,
0.893914474519227313754950317785, 1.91986023388424447428465672186, 3.03862808860091806089384428942, 3.85464774016382354694199052601, 5.03452139139164025086740393084, 5.71584198271652716078803442801, 6.70870919036987273729668193862, 7.23360785145816925042800614774, 8.245525450570863198102638050201, 8.542222737489697285052910572286
