| L(s) = 1 | − i·3-s + 5-s − 3i·7-s + 2·9-s − 2·11-s + (−3 − 2i)13-s − i·15-s + 3·17-s − 3·21-s + 6·23-s − 4·25-s − 5i·27-s − 6i·29-s + 2i·33-s − 3i·35-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 0.447·5-s − 1.13i·7-s + 0.666·9-s − 0.603·11-s + (−0.832 − 0.554i)13-s − 0.258i·15-s + 0.727·17-s − 0.654·21-s + 1.25·23-s − 0.800·25-s − 0.962i·27-s − 1.11i·29-s + 0.348i·33-s − 0.507i·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.196 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.196 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.11735 - 0.916013i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.11735 - 0.916013i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + (3 + 2i)T \) |
| good | 3 | \( 1 + iT - 3T^{2} \) |
| 5 | \( 1 - T + 5T^{2} \) |
| 7 | \( 1 + 3iT - 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 17 | \( 1 - 3T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 3T + 37T^{2} \) |
| 41 | \( 1 + 10iT - 41T^{2} \) |
| 43 | \( 1 - 9iT - 43T^{2} \) |
| 47 | \( 1 - 7iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 10T + 59T^{2} \) |
| 61 | \( 1 - 10iT - 61T^{2} \) |
| 67 | \( 1 - 12T + 67T^{2} \) |
| 71 | \( 1 - 5iT - 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 16T + 83T^{2} \) |
| 89 | \( 1 - 4iT - 89T^{2} \) |
| 97 | \( 1 - 18iT - 97T^{2} \) |
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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
−10.83228872062438302440557434694, −10.14138038224859624964234125400, −9.430494786937664535759407668439, −7.76497461112482200405099189371, −7.53689314953457584073246640950, −6.41491769231875701474624915748, −5.23526122000263984690340009282, −4.08821981060230659903242129669, −2.57871025879883010529356440435, −1.00743657042411183446450197924,
1.98700077270846129269061733720, 3.28019333367342851958767060337, 4.81797216331020672173860035645, 5.42281091134998164623675271986, 6.68219372843844639588494615195, 7.74632184836783213897782844514, 8.955636128101187260718184376317, 9.605097811540624682727093877165, 10.32442858055002501858250137540, 11.37168840793787288233172923128
