L(s) = 1 | + (0.986 + 1.01i)2-s + (0.718 + 0.718i)3-s + (−0.0523 + 1.99i)4-s + (2.02 − 2.02i)5-s + (−0.0188 + 1.43i)6-s + 0.407i·7-s + (−2.07 + 1.91i)8-s − 1.96i·9-s + (4.04 + 0.0529i)10-s + (−1.76 + 1.76i)11-s + (−1.47 + 1.39i)12-s + (−0.707 − 0.707i)13-s + (−0.413 + 0.402i)14-s + 2.90·15-s + (−3.99 − 0.209i)16-s − 4.09·17-s + ⋯ |
L(s) = 1 | + (0.697 + 0.716i)2-s + (0.414 + 0.414i)3-s + (−0.0261 + 0.999i)4-s + (0.903 − 0.903i)5-s + (−0.00767 + 0.586i)6-s + 0.154i·7-s + (−0.734 + 0.678i)8-s − 0.656i·9-s + (1.27 + 0.0167i)10-s + (−0.530 + 0.530i)11-s + (−0.425 + 0.403i)12-s + (−0.196 − 0.196i)13-s + (−0.110 + 0.107i)14-s + 0.749·15-s + (−0.998 − 0.0523i)16-s − 0.991·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.430 - 0.902i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.430 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.67803 + 1.05873i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.67803 + 1.05873i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.986 - 1.01i)T \) |
| 13 | \( 1 + (0.707 + 0.707i)T \) |
good | 3 | \( 1 + (-0.718 - 0.718i)T + 3iT^{2} \) |
| 5 | \( 1 + (-2.02 + 2.02i)T - 5iT^{2} \) |
| 7 | \( 1 - 0.407iT - 7T^{2} \) |
| 11 | \( 1 + (1.76 - 1.76i)T - 11iT^{2} \) |
| 17 | \( 1 + 4.09T + 17T^{2} \) |
| 19 | \( 1 + (2.13 + 2.13i)T + 19iT^{2} \) |
| 23 | \( 1 - 2.42iT - 23T^{2} \) |
| 29 | \( 1 + (-2.51 - 2.51i)T + 29iT^{2} \) |
| 31 | \( 1 - 0.736T + 31T^{2} \) |
| 37 | \( 1 + (-1.13 + 1.13i)T - 37iT^{2} \) |
| 41 | \( 1 + 10.6iT - 41T^{2} \) |
| 43 | \( 1 + (0.341 - 0.341i)T - 43iT^{2} \) |
| 47 | \( 1 - 11.4T + 47T^{2} \) |
| 53 | \( 1 + (6.62 - 6.62i)T - 53iT^{2} \) |
| 59 | \( 1 + (9.01 - 9.01i)T - 59iT^{2} \) |
| 61 | \( 1 + (2.36 + 2.36i)T + 61iT^{2} \) |
| 67 | \( 1 + (-0.526 - 0.526i)T + 67iT^{2} \) |
| 71 | \( 1 - 7.15iT - 71T^{2} \) |
| 73 | \( 1 - 5.01iT - 73T^{2} \) |
| 79 | \( 1 - 9.29T + 79T^{2} \) |
| 83 | \( 1 + (10.8 + 10.8i)T + 83iT^{2} \) |
| 89 | \( 1 + 11.8iT - 89T^{2} \) |
| 97 | \( 1 - 18.7T + 97T^{2} \) |
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
−12.78023716284405065548694840940, −12.05153746564778327035020707963, −10.49846706730961714610273179013, −9.132090591810243924606254449126, −8.857666466447713175393401848882, −7.39378400356823674547988306286, −6.17290796140356723979294682004, −5.14394552355198416326221890581, −4.17234299138492228428824001302, −2.52841996628236417998525214827,
2.06370230788074642376650273351, 2.89689492437296955638655692324, 4.59471797378318886663878896962, 5.95414752017975700877798072951, 6.80534730328314772793401283975, 8.246426318053333839244057813048, 9.581160701277001433824528982387, 10.58077426189943406053643653238, 11.02624608882342590076061499114, 12.41327857081283878576774095658