| L(s) = 1 | + 5.61e4i·2-s + 5.48e7i·3-s + 5.44e9·4-s − 3.07e12·6-s − 7.38e13i·7-s + 7.87e14i·8-s + 2.54e15·9-s + 3.54e16·11-s + 2.98e17i·12-s − 1.27e18i·13-s + 4.14e18·14-s + 2.54e18·16-s − 1.95e20i·17-s + 1.43e20i·18-s − 9.67e20·19-s + ⋯ |
| L(s) = 1 | + 0.605i·2-s + 0.735i·3-s + 0.633·4-s − 0.445·6-s − 0.839i·7-s + 0.989i·8-s + 0.458·9-s + 0.232·11-s + 0.466i·12-s − 0.530i·13-s + 0.508·14-s + 0.0345·16-s − 0.974i·17-s + 0.277i·18-s − 0.769·19-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
| \(L(17)\) |
\(\approx\) |
\(1.447093713\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.447093713\) |
| \(L(\frac{35}{2})\) |
|
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 | \( 1 - 5.61e4iT - 8.58e9T^{2} \) |
| 3 | \( 1 - 5.48e7iT - 5.55e15T^{2} \) |
| 7 | \( 1 + 7.38e13iT - 7.73e27T^{2} \) |
| 11 | \( 1 - 3.54e16T + 2.32e34T^{2} \) |
| 13 | \( 1 + 1.27e18iT - 5.75e36T^{2} \) |
| 17 | \( 1 + 1.95e20iT - 4.02e40T^{2} \) |
| 19 | \( 1 + 9.67e20T + 1.58e42T^{2} \) |
| 23 | \( 1 + 3.06e22iT - 8.65e44T^{2} \) |
| 29 | \( 1 - 1.16e23T + 1.81e48T^{2} \) |
| 31 | \( 1 + 3.89e23T + 1.64e49T^{2} \) |
| 37 | \( 1 - 2.73e25iT - 5.63e51T^{2} \) |
| 41 | \( 1 + 7.27e26T + 1.66e53T^{2} \) |
| 43 | \( 1 + 1.74e27iT - 8.02e53T^{2} \) |
| 47 | \( 1 + 2.14e26iT - 1.51e55T^{2} \) |
| 53 | \( 1 + 1.96e28iT - 7.96e56T^{2} \) |
| 59 | \( 1 + 2.54e29T + 2.74e58T^{2} \) |
| 61 | \( 1 + 2.81e29T + 8.23e58T^{2} \) |
| 67 | \( 1 - 8.18e29iT - 1.82e60T^{2} \) |
| 71 | \( 1 + 3.59e30T + 1.23e61T^{2} \) |
| 73 | \( 1 + 2.19e30iT - 3.08e61T^{2} \) |
| 79 | \( 1 + 3.90e31T + 4.18e62T^{2} \) |
| 83 | \( 1 - 4.98e30iT - 2.13e63T^{2} \) |
| 89 | \( 1 - 1.23e32T + 2.13e64T^{2} \) |
| 97 | \( 1 + 1.97e32iT - 3.65e65T^{2} \) |
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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
−10.79133018021210618845663114804, −10.15390610683763456325313546656, −8.664231380179052181132843219214, −7.38293509836235942169647094152, −6.61528241156618843063971116741, −5.22097651856664745247847973377, −4.20678828739778866822468850384, −2.97838017256158632538630422168, −1.61345755070393448723324211657, −0.23110570791069817622077515623,
1.40070095400281620948109180825, 1.80417515349354200102530935654, 2.95115593749972834070391642375, 4.22338755752969121520843759523, 5.96923443110721633318963183521, 6.76837579221050932161081470156, 7.922562592186172413638976162816, 9.307663355624763822238382769278, 10.53446428399000821000002709248, 11.70906214610517552712800461603
