| L(s) = 1 | + (−55 − 10i)5-s + 158i·7-s − 148·11-s + 684i·13-s + 2.04e3i·17-s + 2.22e3·19-s − 1.24e3i·23-s + (2.92e3 + 1.10e3i)25-s − 270·29-s + 2.04e3·31-s + (1.58e3 − 8.69e3i)35-s + 4.37e3i·37-s + 2.39e3·41-s − 2.29e3i·43-s + 1.06e4i·47-s + ⋯ |
| L(s) = 1 | + (−0.983 − 0.178i)5-s + 1.21i·7-s − 0.368·11-s + 1.12i·13-s + 1.71i·17-s + 1.41·19-s − 0.491i·23-s + (0.936 + 0.352i)25-s − 0.0596·29-s + 0.382·31-s + (0.218 − 1.19i)35-s + 0.525i·37-s + 0.222·41-s − 0.189i·43-s + 0.705i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 - 0.178i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.983 - 0.178i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.123461028\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.123461028\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (55 + 10i)T \) |
| good | 7 | \( 1 - 158iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 148T + 1.61e5T^{2} \) |
| 13 | \( 1 - 684iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 2.04e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 2.22e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.24e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 270T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.04e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 4.37e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 2.39e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.29e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 1.06e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 2.96e3iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 3.97e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.22e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.20e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 4.24e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.01e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 3.52e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.78e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 8.52e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 9.72e4iT - 8.58e9T^{2} \) |
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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
−9.997237747398641560083973834876, −8.985464609204727567999621514017, −8.423186770586250797940167920172, −7.58788739672034986071780418237, −6.52255924616614531058194645684, −5.58798640295237629456379136168, −4.59151064934415568329109083736, −3.62508818867320182130947131949, −2.50743321589217937955076141410, −1.28083330032161261943830772669,
0.30352029737504893863687685317, 0.940046955472601381627138948055, 2.86044218469418759283312630677, 3.55316429666574274638435600366, 4.65162063057916416811697579632, 5.48695229653567532681706656070, 7.00001335816922293298812854567, 7.47366335870589045784942961799, 8.077400479934443942902778531018, 9.355668036513258762201909828199
