L(s) = 1 | + (2.29 + 1.32i)2-s + (−0.335 − 1.25i)3-s + (2.51 + 4.34i)4-s + (0.889 − 3.31i)6-s + (−0.0561 − 0.0972i)7-s + 8.00i·8-s + (1.14 − 0.658i)9-s + (0.479 + 1.78i)11-s + (4.60 − 4.60i)12-s + (−2.96 − 2.05i)13-s − 0.297i·14-s + (−5.58 + 9.67i)16-s + (2.63 + 0.706i)17-s + 3.49·18-s + (−6.72 − 1.80i)19-s + ⋯ |
L(s) = 1 | + (1.62 + 0.936i)2-s + (−0.193 − 0.723i)3-s + (1.25 + 2.17i)4-s + (0.363 − 1.35i)6-s + (−0.0212 − 0.0367i)7-s + 2.83i·8-s + (0.380 − 0.219i)9-s + (0.144 + 0.539i)11-s + (1.32 − 1.32i)12-s + (−0.821 − 0.569i)13-s − 0.0795i·14-s + (−1.39 + 2.41i)16-s + (0.639 + 0.171i)17-s + 0.823·18-s + (−1.54 − 0.413i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.573 - 0.819i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.573 - 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.59448 + 1.35114i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.59448 + 1.35114i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 13 | \( 1 + (2.96 + 2.05i)T \) |
good | 2 | \( 1 + (-2.29 - 1.32i)T + (1 + 1.73i)T^{2} \) |
| 3 | \( 1 + (0.335 + 1.25i)T + (-2.59 + 1.5i)T^{2} \) |
| 7 | \( 1 + (0.0561 + 0.0972i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.479 - 1.78i)T + (-9.52 + 5.5i)T^{2} \) |
| 17 | \( 1 + (-2.63 - 0.706i)T + (14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (6.72 + 1.80i)T + (16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + (-3.10 + 0.831i)T + (19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (4.03 + 2.32i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (0.624 + 0.624i)T + 31iT^{2} \) |
| 37 | \( 1 + (-0.737 + 1.27i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (5.24 - 1.40i)T + (35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (1.00 - 3.76i)T + (-37.2 - 21.5i)T^{2} \) |
| 47 | \( 1 - 0.345T + 47T^{2} \) |
| 53 | \( 1 + (-3.59 + 3.59i)T - 53iT^{2} \) |
| 59 | \( 1 + (-0.332 + 1.24i)T + (-51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (-1.39 - 2.41i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (0.124 + 0.0721i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (1.41 - 5.28i)T + (-61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 - 9.06iT - 73T^{2} \) |
| 79 | \( 1 - 15.1iT - 79T^{2} \) |
| 83 | \( 1 + 8.53T + 83T^{2} \) |
| 89 | \( 1 + (-0.549 + 0.147i)T + (77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (-12.9 + 7.48i)T + (48.5 - 84.0i)T^{2} \) |
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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
−12.26212916516387536333423550960, −11.37766508490514828178797402865, −9.994760441520722488572183242200, −8.386442346570276706045343941892, −7.37089060854196002483894344660, −6.84235858952263748203845962344, −5.89158232817749341783011476407, −4.84372533243697722965582293548, −3.83064283062783203623128888183, −2.34342666056408033567308632419,
1.90585133946080632581520320306, 3.34460376520069752825403061805, 4.30980954427213889450494985272, 5.08525280134262425069293461876, 6.07183217465086733493048799063, 7.24733371307160906345677144813, 9.086024401267612100278392796827, 10.16690695808122627077795109313, 10.70160877178665107816625731301, 11.57085315578815194888651281516