| L(s) = 1 | + 4i·2-s + 19.2i·3-s − 16·4-s − 76.8·6-s − 51.1i·7-s − 64i·8-s − 126.·9-s + 494.·11-s − 307. i·12-s + 169i·13-s + 204.·14-s + 256·16-s − 2.36e3i·17-s − 506. i·18-s + 699.·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 1.23i·3-s − 0.5·4-s − 0.872·6-s − 0.394i·7-s − 0.353i·8-s − 0.520·9-s + 1.23·11-s − 0.616i·12-s + 0.277i·13-s + 0.279·14-s + 0.250·16-s − 1.98i·17-s − 0.368i·18-s + 0.444·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.8228182837\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8228182837\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4iT \) |
| 5 | \( 1 \) |
| 13 | \( 1 - 169iT \) |
| good | 3 | \( 1 - 19.2iT - 243T^{2} \) |
| 7 | \( 1 + 51.1iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 494.T + 1.61e5T^{2} \) |
| 17 | \( 1 + 2.36e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 699.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.80e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 2.26e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 9.08e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 4.48e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 1.24e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.25e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 3.60e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 7.54e3iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 5.00e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.47e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.37e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 7.60e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.16e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 1.03e5T + 3.07e9T^{2} \) |
| 83 | \( 1 - 8.03e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 5.93e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 7.34e4iT - 8.58e9T^{2} \) |
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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
−10.03594608384013982426297663478, −9.281568845852005915115115507881, −9.100477367289480306613518358409, −7.49701310697984778208323927726, −7.01463455501888753939298579420, −5.69599434135675158464679069133, −4.91076561495714424923936568113, −4.02481829338212262406214068930, −3.27596891446237640375953424171, −1.33035435032325176047832885419,
0.17792521344228198541589631241, 1.45490348744143616148447274799, 1.93858970345071507794288773128, 3.32659167346692373838370124474, 4.31966823979743706088062782619, 5.77644640849140991196864230450, 6.47624647882894459242544956580, 7.45490496960687785093462692422, 8.456175823487053119890965243047, 9.028335535635023629845334942652
