L(s) = 1 | + 1.82e4i·2-s − 3.67e7i·3-s + 8.25e9·4-s + 6.70e11·6-s + 1.51e13i·7-s + 3.07e14i·8-s + 4.20e15·9-s − 2.61e17·11-s − 3.03e17i·12-s − 3.70e17i·13-s − 2.76e17·14-s + 6.53e19·16-s + 3.71e20i·17-s + 7.68e19i·18-s − 1.42e21·19-s + ⋯ |
L(s) = 1 | + 0.197i·2-s − 0.492i·3-s + 0.961·4-s + 0.0970·6-s + 0.172i·7-s + 0.386i·8-s + 0.757·9-s − 1.71·11-s − 0.473i·12-s − 0.154i·13-s − 0.0339·14-s + 0.885·16-s + 1.85i·17-s + 0.149i·18-s − 1.13·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(34-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+33/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(17)\) |
\(\approx\) |
\(2.351067262\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.351067262\) |
\(L(\frac{35}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
good | 2 | \( 1 - 1.82e4iT - 8.58e9T^{2} \) |
| 3 | \( 1 + 3.67e7iT - 5.55e15T^{2} \) |
| 7 | \( 1 - 1.51e13iT - 7.73e27T^{2} \) |
| 11 | \( 1 + 2.61e17T + 2.32e34T^{2} \) |
| 13 | \( 1 + 3.70e17iT - 5.75e36T^{2} \) |
| 17 | \( 1 - 3.71e20iT - 4.02e40T^{2} \) |
| 19 | \( 1 + 1.42e21T + 1.58e42T^{2} \) |
| 23 | \( 1 + 3.26e22iT - 8.65e44T^{2} \) |
| 29 | \( 1 + 6.41e23T + 1.81e48T^{2} \) |
| 31 | \( 1 + 1.44e23T + 1.64e49T^{2} \) |
| 37 | \( 1 + 1.34e26iT - 5.63e51T^{2} \) |
| 41 | \( 1 - 6.37e26T + 1.66e53T^{2} \) |
| 43 | \( 1 - 3.37e26iT - 8.02e53T^{2} \) |
| 47 | \( 1 + 3.12e27iT - 1.51e55T^{2} \) |
| 53 | \( 1 + 4.06e28iT - 7.96e56T^{2} \) |
| 59 | \( 1 - 8.77e28T + 2.74e58T^{2} \) |
| 61 | \( 1 + 1.84e28T + 8.23e58T^{2} \) |
| 67 | \( 1 + 7.69e29iT - 1.82e60T^{2} \) |
| 71 | \( 1 - 2.72e30T + 1.23e61T^{2} \) |
| 73 | \( 1 - 4.44e30iT - 3.08e61T^{2} \) |
| 79 | \( 1 - 1.16e31T + 4.18e62T^{2} \) |
| 83 | \( 1 + 4.84e31iT - 2.13e63T^{2} \) |
| 89 | \( 1 + 1.18e32T + 2.13e64T^{2} \) |
| 97 | \( 1 - 2.71e31iT - 3.65e65T^{2} \) |
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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
−10.86161778338240107546794127209, −10.33156700330633895827374410537, −8.357886747343838839807706095498, −7.58982788660344833436748391601, −6.48688264400321165991913597139, −5.55512141917615913114761367192, −4.01054478804020566836204802694, −2.42784486307895581913151133398, −1.91358164485269753717791077145, −0.45357142848253631996093133588,
0.922407971207842133292003437468, 2.23095275252770717189151493754, 3.07188432503583587156760014192, 4.45802200376022608896813396494, 5.56057458411127751188726964971, 7.02363439575900582732702836769, 7.77433627010861506131763894815, 9.519152596400126870536759865828, 10.44537907562767273870896710708, 11.26098286883180172114558723789