| L(s) = 1 | + 0.741i·2-s + 1.44·4-s + (2.33 + 1.24i)7-s + 2.55i·8-s − 1.41i·11-s − 5.54i·13-s + (−0.923 + 1.73i)14-s + 1.00·16-s + 6.07·17-s − 7.12i·19-s + 1.04·22-s + 4.78i·23-s + 4.11·26-s + (3.38 + 1.80i)28-s − 5.51i·29-s + ⋯ |
| L(s) = 1 | + 0.524i·2-s + 0.724·4-s + (0.882 + 0.470i)7-s + 0.904i·8-s − 0.426i·11-s − 1.53i·13-s + (−0.246 + 0.462i)14-s + 0.250·16-s + 1.47·17-s − 1.63i·19-s + 0.223·22-s + 0.997i·23-s + 0.806·26-s + (0.639 + 0.341i)28-s − 1.02i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 - 0.448i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.893 - 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.462111418\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.462111418\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2.33 - 1.24i)T \) |
| good | 2 | \( 1 - 0.741iT - 2T^{2} \) |
| 11 | \( 1 + 1.41iT - 11T^{2} \) |
| 13 | \( 1 + 5.54iT - 13T^{2} \) |
| 17 | \( 1 - 6.07T + 17T^{2} \) |
| 19 | \( 1 + 7.12iT - 19T^{2} \) |
| 23 | \( 1 - 4.78iT - 23T^{2} \) |
| 29 | \( 1 + 5.51iT - 29T^{2} \) |
| 31 | \( 1 + 1.30iT - 31T^{2} \) |
| 37 | \( 1 - 2.57T + 37T^{2} \) |
| 41 | \( 1 + 5.95T + 41T^{2} \) |
| 43 | \( 1 - 6.76T + 43T^{2} \) |
| 47 | \( 1 + 7.83T + 47T^{2} \) |
| 53 | \( 1 - 9.90iT - 53T^{2} \) |
| 59 | \( 1 + 1.84T + 59T^{2} \) |
| 61 | \( 1 - 11.6iT - 61T^{2} \) |
| 67 | \( 1 + 7.23T + 67T^{2} \) |
| 71 | \( 1 + 8.34iT - 71T^{2} \) |
| 73 | \( 1 - 0.559iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 7.83T + 83T^{2} \) |
| 89 | \( 1 + 12.3T + 89T^{2} \) |
| 97 | \( 1 - 5.54iT - 97T^{2} \) |
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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
−9.383648467399214043292264724544, −8.395850159048641924735631568950, −7.78600038255869185449632568733, −7.32322241198442894114925223670, −6.02508812972066443457120359990, −5.59506887268074742486725296311, −4.80610682226399781967711017135, −3.26009840699924487055416835720, −2.51895244552135680764495115159, −1.12579849989770380855479460286,
1.36190528049118673031575099728, 1.96673831081274889425166297714, 3.34546303315945139512828522926, 4.14521522209593991407085186388, 5.13657354260787046464238800829, 6.24804042213421560895520398153, 7.00121558925202350947991181004, 7.73603497396428193378231400224, 8.485682030591863959832840041961, 9.686631413804544518485883440825
