L(s) = 1 | − 1.90i·2-s + i·3-s − 1.62·4-s + 1.90·6-s + 4.42i·7-s − 0.719i·8-s − 9-s + 11-s − 1.62i·12-s − 0.622i·13-s + 8.42·14-s − 4.61·16-s + 5.18i·17-s + 1.90i·18-s − 7.05·19-s + ⋯ |
L(s) = 1 | − 1.34i·2-s + 0.577i·3-s − 0.811·4-s + 0.776·6-s + 1.67i·7-s − 0.254i·8-s − 0.333·9-s + 0.301·11-s − 0.468i·12-s − 0.172i·13-s + 2.25·14-s − 1.15·16-s + 1.25i·17-s + 0.448i·18-s − 1.61·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.22912 + 0.290157i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.22912 + 0.290157i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 2 | \( 1 + 1.90iT - 2T^{2} \) |
| 7 | \( 1 - 4.42iT - 7T^{2} \) |
| 13 | \( 1 + 0.622iT - 13T^{2} \) |
| 17 | \( 1 - 5.18iT - 17T^{2} \) |
| 19 | \( 1 + 7.05T + 19T^{2} \) |
| 23 | \( 1 - 8.85iT - 23T^{2} \) |
| 29 | \( 1 - 7.80T + 29T^{2} \) |
| 31 | \( 1 - 2.75T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 0.193T + 41T^{2} \) |
| 43 | \( 1 - 5.67iT - 43T^{2} \) |
| 47 | \( 1 - 2.75iT - 47T^{2} \) |
| 53 | \( 1 + 10.8iT - 53T^{2} \) |
| 59 | \( 1 - 4.85T + 59T^{2} \) |
| 61 | \( 1 - 6.85T + 61T^{2} \) |
| 67 | \( 1 - 1.24iT - 67T^{2} \) |
| 71 | \( 1 - 2.75T + 71T^{2} \) |
| 73 | \( 1 - 4.23iT - 73T^{2} \) |
| 79 | \( 1 + 8.56T + 79T^{2} \) |
| 83 | \( 1 - 0.133iT - 83T^{2} \) |
| 89 | \( 1 + 5.61T + 89T^{2} \) |
| 97 | \( 1 + 7.24iT - 97T^{2} \) |
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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
−10.24197673579170248946118042968, −9.739016674954097129569025345905, −8.770720173653183306010615550923, −8.312011013133390768081120632367, −6.55819994393556537648240276035, −5.76845360406980463442903253493, −4.62968548506691528379060573417, −3.62709873265667025876242237144, −2.66738533295421041852669406306, −1.71452847983268594848801885341,
0.61630173451540393108419477284, 2.47068811344536406231443898767, 4.18583498199407337787606462020, 4.84477184131292186806258237260, 6.28129337596084193926299431910, 6.76421110771962118825082572381, 7.32895975383304295692938745382, 8.241991683965615809637707023092, 8.866396960897427004998479489509, 10.21016994992763427718016142471