| L(s) = 1 | + 2i·2-s − 3i·3-s − 4·4-s + 6·6-s − 4i·7-s − 8i·8-s − 9·9-s − 48·11-s + 12i·12-s − 2i·13-s + 8·14-s + 16·16-s − 114i·17-s − 18i·18-s − 140·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + 0.408·6-s − 0.215i·7-s − 0.353i·8-s − 0.333·9-s − 1.31·11-s + 0.288i·12-s − 0.0426i·13-s + 0.152·14-s + 0.250·16-s − 1.62i·17-s − 0.235i·18-s − 1.69·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.311319 - 0.503725i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.311319 - 0.503725i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2iT \) |
| 3 | \( 1 + 3iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 4iT - 343T^{2} \) |
| 11 | \( 1 + 48T + 1.33e3T^{2} \) |
| 13 | \( 1 + 2iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 114iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 140T + 6.85e3T^{2} \) |
| 23 | \( 1 + 72iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 210T + 2.43e4T^{2} \) |
| 31 | \( 1 - 272T + 2.97e4T^{2} \) |
| 37 | \( 1 + 334iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 198T + 6.89e4T^{2} \) |
| 43 | \( 1 - 268iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 216iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 78iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 240T + 2.05e5T^{2} \) |
| 61 | \( 1 - 302T + 2.26e5T^{2} \) |
| 67 | \( 1 - 596iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 768T + 3.57e5T^{2} \) |
| 73 | \( 1 - 478iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 640T + 4.93e5T^{2} \) |
| 83 | \( 1 - 348iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 210T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.53e3iT - 9.12e5T^{2} \) |
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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
−12.56129776480021685096843606329, −11.21083133443429788769480921893, −10.12795762288647170570320878774, −8.823280012398518663406337505394, −7.82624810855910269559865874379, −6.96407135356038098304008064783, −5.76280917360728651188354811322, −4.54470384811990768391320096075, −2.58079993469889873831825000244, −0.26970457103126915749408202189,
2.15063714118031055240266891461, 3.64809476994345000178108283069, 4.89803569275988450126733203883, 6.13617407268324372371099619580, 8.008723332242600693263282145555, 8.815830045508566178127693547766, 10.22576905109508936003417735427, 10.59147494896833370313373735651, 11.78821021563569664780997133751, 12.86979543112994399977079057624
