L(s) = 1 | + (0.707 − 0.707i)2-s + (−0.541 + 0.541i)3-s − 1.00i·4-s + (2.23 + 0.158i)5-s + 0.765i·6-s + (−2.14 − 1.55i)7-s + (−0.707 − 0.707i)8-s + 2.41i·9-s + (1.68 − 1.46i)10-s − 2.82·11-s + (0.541 + 0.541i)12-s + (−2.83 + 2.83i)13-s + (−2.61 + 0.414i)14-s + (−1.29 + 1.12i)15-s − 1.00·16-s + (−1.53 − 1.53i)17-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s + (−0.312 + 0.312i)3-s − 0.500i·4-s + (0.997 + 0.0708i)5-s + 0.312i·6-s + (−0.809 − 0.587i)7-s + (−0.250 − 0.250i)8-s + 0.804i·9-s + (0.534 − 0.463i)10-s − 0.852·11-s + (0.156 + 0.156i)12-s + (−0.786 + 0.786i)13-s + (−0.698 + 0.110i)14-s + (−0.333 + 0.289i)15-s − 0.250·16-s + (−0.371 − 0.371i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.892 + 0.450i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.892 + 0.450i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.05475 - 0.251038i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.05475 - 0.251038i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 5 | \( 1 + (-2.23 - 0.158i)T \) |
| 7 | \( 1 + (2.14 + 1.55i)T \) |
good | 3 | \( 1 + (0.541 - 0.541i)T - 3iT^{2} \) |
| 11 | \( 1 + 2.82T + 11T^{2} \) |
| 13 | \( 1 + (2.83 - 2.83i)T - 13iT^{2} \) |
| 17 | \( 1 + (1.53 + 1.53i)T + 17iT^{2} \) |
| 19 | \( 1 - 7.07T + 19T^{2} \) |
| 23 | \( 1 + (2.41 + 2.41i)T + 23iT^{2} \) |
| 29 | \( 1 + 4.82iT - 29T^{2} \) |
| 31 | \( 1 - 3.69iT - 31T^{2} \) |
| 37 | \( 1 + (-5.41 + 5.41i)T - 37iT^{2} \) |
| 41 | \( 1 + 1.53iT - 41T^{2} \) |
| 43 | \( 1 + (-4 - 4i)T + 43iT^{2} \) |
| 47 | \( 1 + (-2.61 - 2.61i)T + 47iT^{2} \) |
| 53 | \( 1 + (-0.242 - 0.242i)T + 53iT^{2} \) |
| 59 | \( 1 + 3.82T + 59T^{2} \) |
| 61 | \( 1 + 10.3iT - 61T^{2} \) |
| 67 | \( 1 + (6.48 - 6.48i)T - 67iT^{2} \) |
| 71 | \( 1 + 3.41T + 71T^{2} \) |
| 73 | \( 1 + (4.77 - 4.77i)T - 73iT^{2} \) |
| 79 | \( 1 - 9.07iT - 79T^{2} \) |
| 83 | \( 1 + (-5.45 + 5.45i)T - 83iT^{2} \) |
| 89 | \( 1 + 16.9T + 89T^{2} \) |
| 97 | \( 1 + (-11.0 - 11.0i)T + 97iT^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.15802908858313309159059711732, −13.62659627269665875682464719362, −12.60698917307634075801568521057, −11.19823815161699470453422170130, −10.15427252996612295131584648425, −9.514678202187531583154500937908, −7.32436710155711467761048451089, −5.83330329117595111325775970775, −4.64317506760503177735883080044, −2.58592318576753850291609497310,
2.95982849167023176394994439581, 5.34775913171713685437555407601, 6.12279359553086067769335308622, 7.44241469324502942312229552703, 9.149232712787918132247877102547, 10.12843959117082993345242172766, 11.91960979904304456243877324997, 12.80665205047443580777822070152, 13.53174443633913318970367956029, 14.86889951754874331666181666986