| L(s) = 1 | − 10.1i·2-s + 24.5i·3-s − 70.0·4-s + 247.·6-s − 72.1i·7-s + 384. i·8-s − 358.·9-s + 127.·11-s − 1.71e3i·12-s − 169i·13-s − 728.·14-s + 1.64e3·16-s + 2.15e3i·17-s + 3.61e3i·18-s − 2.72e3·19-s + ⋯ |
| L(s) = 1 | − 1.78i·2-s + 1.57i·3-s − 2.18·4-s + 2.80·6-s − 0.556i·7-s + 2.12i·8-s − 1.47·9-s + 0.317·11-s − 3.44i·12-s − 0.277i·13-s − 0.993·14-s + 1.60·16-s + 1.80i·17-s + 2.63i·18-s − 1.73·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.226203782\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.226203782\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 13 | \( 1 + 169iT \) |
| good | 2 | \( 1 + 10.1iT - 32T^{2} \) |
| 3 | \( 1 - 24.5iT - 243T^{2} \) |
| 7 | \( 1 + 72.1iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 127.T + 1.61e5T^{2} \) |
| 17 | \( 1 - 2.15e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 2.72e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.65e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 5.88e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.36e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 481. iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 7.38e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.01e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 1.63e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 9.06e3iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 2.95e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.64e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.15e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 5.22e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.78e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 8.95e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 7.89e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 1.11e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 7.15e4iT - 8.58e9T^{2} \) |
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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
−10.57127637115679268052131421827, −10.09778378512772204987001551623, −8.935077243303355664054559841755, −8.397743745190988775626943034507, −6.20197205332137580795549425688, −4.65205638026373899382546098118, −4.16259405974190355586257935713, −3.34818940643426416303641401175, −2.02993396173403012153945068077, −0.43031295940129504376659832031,
0.875135445223271934174466190226, 2.50781147978392259357850902122, 4.51924846111782127497741709882, 5.67984795001947724957616893709, 6.50722936244026230425058337375, 7.07262865687862113074382046392, 7.949067440612872588068863616525, 8.692238740307307371459756765837, 9.554444891715790156695952603413, 11.39866621158612150537494877409
