L(s) = 1 | + (1.15 − 1.29i)3-s + (−0.186 + 0.186i)5-s − 7-s + (−0.350 − 2.97i)9-s + (4.29 + 4.29i)11-s + (2.73 − 2.73i)13-s + (0.0267 + 0.455i)15-s − 4.98i·17-s + (3.09 + 3.09i)19-s + (−1.15 + 1.29i)21-s − 3.55i·23-s + 4.93i·25-s + (−4.26 − 2.97i)27-s + (−3.75 − 3.75i)29-s + 6.58i·31-s + ⋯ |
L(s) = 1 | + (0.664 − 0.747i)3-s + (−0.0833 + 0.0833i)5-s − 0.377·7-s + (−0.116 − 0.993i)9-s + (1.29 + 1.29i)11-s + (0.759 − 0.759i)13-s + (0.00691 + 0.117i)15-s − 1.20i·17-s + (0.710 + 0.710i)19-s + (−0.251 + 0.282i)21-s − 0.742i·23-s + 0.986i·25-s + (−0.819 − 0.572i)27-s + (−0.697 − 0.697i)29-s + 1.18i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.562 + 0.826i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.562 + 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.193088926\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.193088926\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.15 + 1.29i)T \) |
| 7 | \( 1 + T \) |
good | 5 | \( 1 + (0.186 - 0.186i)T - 5iT^{2} \) |
| 11 | \( 1 + (-4.29 - 4.29i)T + 11iT^{2} \) |
| 13 | \( 1 + (-2.73 + 2.73i)T - 13iT^{2} \) |
| 17 | \( 1 + 4.98iT - 17T^{2} \) |
| 19 | \( 1 + (-3.09 - 3.09i)T + 19iT^{2} \) |
| 23 | \( 1 + 3.55iT - 23T^{2} \) |
| 29 | \( 1 + (3.75 + 3.75i)T + 29iT^{2} \) |
| 31 | \( 1 - 6.58iT - 31T^{2} \) |
| 37 | \( 1 + (4.82 + 4.82i)T + 37iT^{2} \) |
| 41 | \( 1 - 10.5T + 41T^{2} \) |
| 43 | \( 1 + (-5.79 + 5.79i)T - 43iT^{2} \) |
| 47 | \( 1 - 3.22T + 47T^{2} \) |
| 53 | \( 1 + (-7.95 + 7.95i)T - 53iT^{2} \) |
| 59 | \( 1 + (3.18 + 3.18i)T + 59iT^{2} \) |
| 61 | \( 1 + (3.17 - 3.17i)T - 61iT^{2} \) |
| 67 | \( 1 + (2.39 + 2.39i)T + 67iT^{2} \) |
| 71 | \( 1 + 9.06iT - 71T^{2} \) |
| 73 | \( 1 + 2.32iT - 73T^{2} \) |
| 79 | \( 1 - 6.89iT - 79T^{2} \) |
| 83 | \( 1 + (1.12 - 1.12i)T - 83iT^{2} \) |
| 89 | \( 1 + 7.81T + 89T^{2} \) |
| 97 | \( 1 - 12.6T + 97T^{2} \) |
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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
−9.273029009333226396515186067259, −8.861347356328971670130143361391, −7.60428515884893318412820350840, −7.23156501441458400973822063675, −6.39186753746164811786552203780, −5.43408663510198452717356025625, −4.05557009982673056239672808063, −3.33149400651521216693704035593, −2.16293055121941755522306236473, −0.996194361647919915964886388948,
1.33344035632090221840388949843, 2.82976370103777237003645945633, 3.82360355049864267568046138324, 4.21567751444345321828401674377, 5.69703313098187374873306465699, 6.27986019803480978462276509616, 7.42515588027208263172727402797, 8.400001174195606327103100208862, 9.045467571759406032625866230380, 9.424663955049399559465823482129