L(s) = 1 | + (0.707 + 0.707i)2-s + (1.73 − 0.0456i)3-s + 1.00i·4-s + (−1.54 − 1.62i)5-s + (1.25 + 1.19i)6-s + (0.528 − 0.528i)7-s + (−0.707 + 0.707i)8-s + (2.99 − 0.157i)9-s + (0.0565 − 2.23i)10-s − 4.60i·11-s + (0.0456 + 1.73i)12-s + (4.73 + 4.73i)13-s + 0.746·14-s + (−2.74 − 2.73i)15-s − 1.00·16-s + (3.32 + 3.32i)17-s + ⋯ |
L(s) = 1 | + (0.499 + 0.499i)2-s + (0.999 − 0.0263i)3-s + 0.500i·4-s + (−0.688 − 0.724i)5-s + (0.512 + 0.486i)6-s + (0.199 − 0.199i)7-s + (−0.250 + 0.250i)8-s + (0.998 − 0.0526i)9-s + (0.0178 − 0.706i)10-s − 1.38i·11-s + (0.0131 + 0.499i)12-s + (1.31 + 1.31i)13-s + 0.199·14-s + (−0.707 − 0.706i)15-s − 0.250·16-s + (0.807 + 0.807i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.230i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 - 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.62423 + 0.306926i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.62423 + 0.306926i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.73 + 0.0456i)T \) |
| 5 | \( 1 + (1.54 + 1.62i)T \) |
| 23 | \( 1 + (-0.707 + 0.707i)T \) |
good | 7 | \( 1 + (-0.528 + 0.528i)T - 7iT^{2} \) |
| 11 | \( 1 + 4.60iT - 11T^{2} \) |
| 13 | \( 1 + (-4.73 - 4.73i)T + 13iT^{2} \) |
| 17 | \( 1 + (-3.32 - 3.32i)T + 17iT^{2} \) |
| 19 | \( 1 + 5.21iT - 19T^{2} \) |
| 29 | \( 1 + 2.50T + 29T^{2} \) |
| 31 | \( 1 - 3.91T + 31T^{2} \) |
| 37 | \( 1 + (7.08 - 7.08i)T - 37iT^{2} \) |
| 41 | \( 1 + 4.23iT - 41T^{2} \) |
| 43 | \( 1 + (7.82 + 7.82i)T + 43iT^{2} \) |
| 47 | \( 1 + (-2.42 - 2.42i)T + 47iT^{2} \) |
| 53 | \( 1 + (-1.66 + 1.66i)T - 53iT^{2} \) |
| 59 | \( 1 + 12.7T + 59T^{2} \) |
| 61 | \( 1 + 1.51T + 61T^{2} \) |
| 67 | \( 1 + (5.25 - 5.25i)T - 67iT^{2} \) |
| 71 | \( 1 + 10.9iT - 71T^{2} \) |
| 73 | \( 1 + (5.55 + 5.55i)T + 73iT^{2} \) |
| 79 | \( 1 - 6.67iT - 79T^{2} \) |
| 83 | \( 1 + (9.68 - 9.68i)T - 83iT^{2} \) |
| 89 | \( 1 + 9.07T + 89T^{2} \) |
| 97 | \( 1 + (-5.69 + 5.69i)T - 97iT^{2} \) |
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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
−10.60036663256269607576378002059, −9.079280108478876468034975302450, −8.677234951170112037696172572858, −8.064311836321997396041186213370, −7.05941021388386369293073564268, −6.10883977021675055766873800016, −4.82382477825755642940301111336, −3.90247469748745760718426958319, −3.24883872152364596014084399989, −1.38044617344558877848079111860,
1.61798587700362273630046200522, 2.97771882124923432373289614028, 3.58896058494079465463549517929, 4.62456110798575484724894545711, 5.86906013313732836205399011071, 7.14330447852196747734027840777, 7.80821008007903883648059766328, 8.632538631541270046356525784794, 9.930613365176005025722994864043, 10.25958000624113411013078457066