| L(s) = 1 | + (0.809 − 0.587i)2-s + (−0.924 + 2.84i)3-s + (0.309 − 0.951i)4-s + (−1.56 + 1.59i)5-s + (0.924 + 2.84i)6-s + 7-s + (−0.309 − 0.951i)8-s + (−4.82 − 3.50i)9-s + (−0.328 + 2.21i)10-s + (−4.39 + 3.19i)11-s + (2.42 + 1.75i)12-s + (2.43 + 1.76i)13-s + (0.809 − 0.587i)14-s + (−3.09 − 5.93i)15-s + (−0.809 − 0.587i)16-s + (−0.726 − 2.23i)17-s + ⋯ |
| L(s) = 1 | + (0.572 − 0.415i)2-s + (−0.534 + 1.64i)3-s + (0.154 − 0.475i)4-s + (−0.700 + 0.714i)5-s + (0.377 + 1.16i)6-s + 0.377·7-s + (−0.109 − 0.336i)8-s + (−1.60 − 1.16i)9-s + (−0.103 + 0.699i)10-s + (−1.32 + 0.962i)11-s + (0.699 + 0.507i)12-s + (0.674 + 0.489i)13-s + (0.216 − 0.157i)14-s + (−0.799 − 1.53i)15-s + (−0.202 − 0.146i)16-s + (−0.176 − 0.542i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.698 - 0.715i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.698 - 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.407660 + 0.968058i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.407660 + 0.968058i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.809 + 0.587i)T \) |
| 5 | \( 1 + (1.56 - 1.59i)T \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 + (0.924 - 2.84i)T + (-2.42 - 1.76i)T^{2} \) |
| 11 | \( 1 + (4.39 - 3.19i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (-2.43 - 1.76i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (0.726 + 2.23i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (0.0911 + 0.280i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (3.77 - 2.74i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (0.883 - 2.71i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-2.79 - 8.60i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-5.91 - 4.29i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-4.21 - 3.06i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - 8.77T + 43T^{2} \) |
| 47 | \( 1 + (-0.140 + 0.432i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-4.22 + 13.0i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (-8.60 - 6.25i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (1.98 - 1.44i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (1.53 + 4.73i)T + (-54.2 + 39.3i)T^{2} \) |
| 71 | \( 1 + (-3.93 + 12.1i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (5.08 - 3.69i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (3.84 - 11.8i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-5.37 - 16.5i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (8.14 - 5.91i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-2.10 + 6.48i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−11.51965517225129397876753975462, −10.92889976606148272051013739062, −10.29069544224965388858197148011, −9.514797145926697751377373981688, −8.172578579365489975158362997490, −6.87277515779205810506272113355, −5.54435384936011012903911095490, −4.67749361402302834441089752521, −3.92311532298210820276962637841, −2.74639590030719731732937677782,
0.63633584755145221761222320352, 2.47410593051331795396572989759, 4.20516334234641898730356738006, 5.69170883933361953251022931222, 5.99351201574710487908347868480, 7.56944726976864067590996515771, 7.907723426499977975485716154105, 8.648443125059946699892442803046, 10.76007591917397956186705322231, 11.44272553997418513764775070820
