| L(s) = 1 | + (−5.19 − 0.160i)3-s + 7.59i·5-s + 7i·7-s + (26.9 + 1.66i)9-s + 0.0298·11-s + 40.1·13-s + (1.21 − 39.4i)15-s + 0.439i·17-s − 0.434i·19-s + (1.12 − 36.3i)21-s − 139.·23-s + 67.2·25-s + (−139. − 12.9i)27-s + 273. i·29-s − 149. i·31-s + ⋯ |
| L(s) = 1 | + (−0.999 − 0.0308i)3-s + 0.679i·5-s + 0.377i·7-s + (0.998 + 0.0617i)9-s + 0.000818·11-s + 0.856·13-s + (0.0209 − 0.679i)15-s + 0.00627i·17-s − 0.00524i·19-s + (0.0116 − 0.377i)21-s − 1.26·23-s + 0.538·25-s + (−0.995 − 0.0925i)27-s + 1.75i·29-s − 0.866i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 - 0.526i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.850 - 0.526i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6748491057\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6748491057\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (5.19 + 0.160i)T \) |
| 7 | \( 1 - 7iT \) |
| good | 5 | \( 1 - 7.59iT - 125T^{2} \) |
| 11 | \( 1 - 0.0298T + 1.33e3T^{2} \) |
| 13 | \( 1 - 40.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 0.439iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 0.434iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 139.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 273. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 149. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 290.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 57.4iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 432. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 317.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 147. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 488.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 441.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 440. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 202.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 542.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 325. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 376.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 496. iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 373.T + 9.12e5T^{2} \) |
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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
−11.38357275695982179157592039578, −10.73530897839762561704153208642, −9.916742393017681345688272876851, −8.723753095462760756093871185283, −7.50816643989793906505509607253, −6.50499516857946456263569410177, −5.81020657886740071041404426641, −4.60705481316051231273168628426, −3.26367669070843868022853032456, −1.55055292021744832793741105459,
0.28639312284852547728548079098, 1.60173656027681919162675214890, 3.77173912501392022284535630105, 4.76508000291273527130456529590, 5.79533052120027059607312917300, 6.68965831208792879251507724670, 7.86433360445527900010916471275, 8.895941144430438667207161627552, 10.04045279580609046759765499978, 10.72996827499914924917853023281
