L(s) = 1 | + (−0.707 − 0.707i)2-s + (0.707 + 0.707i)3-s + 1.00i·4-s + (2.14 + 0.631i)5-s − 1.00i·6-s + (2.22 + 2.22i)7-s + (0.707 − 0.707i)8-s + 1.00i·9-s + (−1.07 − 1.96i)10-s + 2.67i·11-s + (−0.707 + 0.707i)12-s + (1.52 + 1.52i)13-s − 3.14i·14-s + (1.07 + 1.96i)15-s − 1.00·16-s + (−1.08 + 1.08i)17-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.499i)2-s + (0.408 + 0.408i)3-s + 0.500i·4-s + (0.959 + 0.282i)5-s − 0.408i·6-s + (0.840 + 0.840i)7-s + (0.250 − 0.250i)8-s + 0.333i·9-s + (−0.338 − 0.620i)10-s + 0.805i·11-s + (−0.204 + 0.204i)12-s + (0.424 + 0.424i)13-s − 0.840i·14-s + (0.276 + 0.506i)15-s − 0.250·16-s + (−0.263 + 0.263i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.611 - 0.791i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.611 - 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.58257 + 0.777642i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.58257 + 0.777642i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 + 0.707i)T \) |
| 3 | \( 1 + (-0.707 - 0.707i)T \) |
| 5 | \( 1 + (-2.14 - 0.631i)T \) |
| 31 | \( 1 + (3.35 + 4.44i)T \) |
good | 7 | \( 1 + (-2.22 - 2.22i)T + 7iT^{2} \) |
| 11 | \( 1 - 2.67iT - 11T^{2} \) |
| 13 | \( 1 + (-1.52 - 1.52i)T + 13iT^{2} \) |
| 17 | \( 1 + (1.08 - 1.08i)T - 17iT^{2} \) |
| 19 | \( 1 - 1.14iT - 19T^{2} \) |
| 23 | \( 1 + (4.52 + 4.52i)T + 23iT^{2} \) |
| 29 | \( 1 + 3.21T + 29T^{2} \) |
| 37 | \( 1 + (0.0599 - 0.0599i)T - 37iT^{2} \) |
| 41 | \( 1 - 1.16T + 41T^{2} \) |
| 43 | \( 1 + (3.62 + 3.62i)T + 43iT^{2} \) |
| 47 | \( 1 + (-0.927 - 0.927i)T + 47iT^{2} \) |
| 53 | \( 1 + (4.04 + 4.04i)T + 53iT^{2} \) |
| 59 | \( 1 + 14.0iT - 59T^{2} \) |
| 61 | \( 1 - 11.1iT - 61T^{2} \) |
| 67 | \( 1 + (-8.98 - 8.98i)T + 67iT^{2} \) |
| 71 | \( 1 - 9.67T + 71T^{2} \) |
| 73 | \( 1 + (-6.81 - 6.81i)T + 73iT^{2} \) |
| 79 | \( 1 - 11.2T + 79T^{2} \) |
| 83 | \( 1 + (11.0 + 11.0i)T + 83iT^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 + (0.00703 + 0.00703i)T + 97iT^{2} \) |
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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
−10.03907119621350796775294687481, −9.455423058692164257323806610104, −8.679956107880467607304994512249, −8.012980331344962606080476796590, −6.85461798189317618060071695513, −5.81483897713558651092233215658, −4.82530398204481639866047554833, −3.75454349387889595065644512229, −2.27839498379140921880647022002, −1.89238788129382059131931648927,
0.985748477278432104158989193797, 1.98910399019993622521518359575, 3.51791241895178874912922124144, 4.87873692337171252207538172560, 5.76041158502753841545903884479, 6.59477338276151228985894648184, 7.59339832901846316249615028190, 8.200484064222350418855181566154, 9.025108119242530736634779437873, 9.737984768736457016195335259716