L(s) = 1 | − 1.90i·2-s − 3i·3-s + 4.36·4-s − 5.71·6-s − 22.9i·7-s − 23.5i·8-s − 9·9-s + 11·11-s − 13.0i·12-s − 66.7i·13-s − 43.6·14-s − 10.0·16-s − 3.45i·17-s + 17.1i·18-s − 78.2·19-s + ⋯ |
L(s) = 1 | − 0.674i·2-s − 0.577i·3-s + 0.545·4-s − 0.389·6-s − 1.23i·7-s − 1.04i·8-s − 0.333·9-s + 0.301·11-s − 0.315i·12-s − 1.42i·13-s − 0.834·14-s − 0.156·16-s − 0.0493i·17-s + 0.224i·18-s − 0.944·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.022597386\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.022597386\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 3iT \) |
| 5 | \( 1 \) |
| 11 | \( 1 - 11T \) |
good | 2 | \( 1 + 1.90iT - 8T^{2} \) |
| 7 | \( 1 + 22.9iT - 343T^{2} \) |
| 13 | \( 1 + 66.7iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 3.45iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 78.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 12.2iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 31.1T + 2.43e4T^{2} \) |
| 31 | \( 1 - 247.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 304. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 29.8T + 6.89e4T^{2} \) |
| 43 | \( 1 + 269. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 225. iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 16.8iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 28.0T + 2.05e5T^{2} \) |
| 61 | \( 1 + 853.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 36.7iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 23.8T + 3.57e5T^{2} \) |
| 73 | \( 1 + 707. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 412.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 552. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 1.49e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.19e3iT - 9.12e5T^{2} \) |
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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
−9.654898789274835125365026796146, −8.303266613297277749813617396556, −7.62577247413108491147666317700, −6.75169491320244976992993318647, −6.10369390968730866312389001709, −4.63987811616725456123901048353, −3.53427035484322211936704981873, −2.65367924798567292715516736935, −1.35308911784514062930252593028, −0.51543036041311502224827911407,
1.87469471772198786620980330762, 2.74450066485110859435988442785, 4.18431131880339789659203921616, 5.14683414185642512479679067533, 6.14747147319619796991084826886, 6.59642025071383759017530636093, 7.79143550542417332935864754273, 8.737131293096025975303248973267, 9.176912140013841068536000832322, 10.29410599905593013257097610385