L(s) = 1 | − 1.40e5i·2-s − 1.29e8i·3-s − 1.11e10·4-s − 1.81e13·6-s + 4.00e13i·7-s + 3.61e14i·8-s − 1.11e16·9-s + 2.48e17·11-s + 1.44e18i·12-s + 3.61e18i·13-s + 5.62e18·14-s − 4.50e19·16-s + 5.03e19i·17-s + 1.56e21i·18-s + 1.19e21·19-s + ⋯ |
L(s) = 1 | − 1.51i·2-s − 1.73i·3-s − 1.29·4-s − 2.62·6-s + 0.455i·7-s + 0.454i·8-s − 1.99·9-s + 1.62·11-s + 2.25i·12-s + 1.50i·13-s + 0.690·14-s − 0.610·16-s + 0.250i·17-s + 3.03i·18-s + 0.952·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(34-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+33/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(17)\) |
\(\approx\) |
\(2.415157962\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.415157962\) |
\(L(\frac{35}{2})\) |
|
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 | \( 1 + 1.40e5iT - 8.58e9T^{2} \) |
| 3 | \( 1 + 1.29e8iT - 5.55e15T^{2} \) |
| 7 | \( 1 - 4.00e13iT - 7.73e27T^{2} \) |
| 11 | \( 1 - 2.48e17T + 2.32e34T^{2} \) |
| 13 | \( 1 - 3.61e18iT - 5.75e36T^{2} \) |
| 17 | \( 1 - 5.03e19iT - 4.02e40T^{2} \) |
| 19 | \( 1 - 1.19e21T + 1.58e42T^{2} \) |
| 23 | \( 1 + 2.35e22iT - 8.65e44T^{2} \) |
| 29 | \( 1 - 2.45e24T + 1.81e48T^{2} \) |
| 31 | \( 1 + 2.88e24T + 1.64e49T^{2} \) |
| 37 | \( 1 + 3.58e25iT - 5.63e51T^{2} \) |
| 41 | \( 1 - 6.61e25T + 1.66e53T^{2} \) |
| 43 | \( 1 - 1.70e27iT - 8.02e53T^{2} \) |
| 47 | \( 1 + 4.53e27iT - 1.51e55T^{2} \) |
| 53 | \( 1 - 1.11e28iT - 7.96e56T^{2} \) |
| 59 | \( 1 + 1.06e29T + 2.74e58T^{2} \) |
| 61 | \( 1 - 9.42e28T + 8.23e58T^{2} \) |
| 67 | \( 1 + 2.59e30iT - 1.82e60T^{2} \) |
| 71 | \( 1 + 4.27e30T + 1.23e61T^{2} \) |
| 73 | \( 1 - 2.23e30iT - 3.08e61T^{2} \) |
| 79 | \( 1 - 2.21e31T + 4.18e62T^{2} \) |
| 83 | \( 1 - 2.60e31iT - 2.13e63T^{2} \) |
| 89 | \( 1 + 1.65e32T + 2.13e64T^{2} \) |
| 97 | \( 1 - 1.37e32iT - 3.65e65T^{2} \) |
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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
−11.21568407473582885367181388452, −9.450191813029330284860941922260, −8.609721414026117835949132759045, −6.99094121951569230307699470709, −6.26574205437165358403274498088, −4.36342571457970988577778261644, −3.06008350091247274887920104310, −2.01948645920868270114599268954, −1.41680394325747864399078736595, −0.65331345698262208407916681585,
0.78174637452262207470667774196, 3.16429017535869065262543833483, 4.11768721007211009668056480222, 5.10165863492477440570301963055, 5.93356373256349564826946120196, 7.26013752275328143859935050216, 8.538489698675482882771897676435, 9.418536034778580244284897410632, 10.42824220602752804048147035337, 11.71702150711283318527861510447