| L(s) = 1 | + (29.1 − 50.5i)3-s + (2.15e3 − 1.24e3i)5-s + (−2.21e3 + 5.95e3i)7-s + (8.13e3 + 1.40e4i)9-s + (−7.19e4 − 4.15e4i)11-s + 8.97e4i·13-s − 1.45e5i·15-s + (−3.29e5 − 1.90e5i)17-s + (−5.17e5 − 8.97e5i)19-s + (2.36e5 + 2.85e5i)21-s + (−1.03e6 + 5.97e5i)23-s + (2.11e6 − 3.65e6i)25-s + 2.09e6·27-s − 3.28e6·29-s + (2.15e5 − 3.73e5i)31-s + ⋯ |
| L(s) = 1 | + (0.208 − 0.360i)3-s + (1.54 − 0.889i)5-s + (−0.348 + 0.937i)7-s + (0.413 + 0.716i)9-s + (−1.48 − 0.855i)11-s + 0.871i·13-s − 0.739i·15-s + (−0.957 − 0.552i)17-s + (−0.911 − 1.57i)19-s + (0.265 + 0.320i)21-s + (−0.771 + 0.445i)23-s + (1.08 − 1.87i)25-s + 0.760·27-s − 0.861·29-s + (0.0419 − 0.0726i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 + 0.289i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.957 + 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.9899746298\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9899746298\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (2.21e3 - 5.95e3i)T \) |
| good | 3 | \( 1 + (-29.1 + 50.5i)T + (-9.84e3 - 1.70e4i)T^{2} \) |
| 5 | \( 1 + (-2.15e3 + 1.24e3i)T + (9.76e5 - 1.69e6i)T^{2} \) |
| 11 | \( 1 + (7.19e4 + 4.15e4i)T + (1.17e9 + 2.04e9i)T^{2} \) |
| 13 | \( 1 - 8.97e4iT - 1.06e10T^{2} \) |
| 17 | \( 1 + (3.29e5 + 1.90e5i)T + (5.92e10 + 1.02e11i)T^{2} \) |
| 19 | \( 1 + (5.17e5 + 8.97e5i)T + (-1.61e11 + 2.79e11i)T^{2} \) |
| 23 | \( 1 + (1.03e6 - 5.97e5i)T + (9.00e11 - 1.55e12i)T^{2} \) |
| 29 | \( 1 + 3.28e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + (-2.15e5 + 3.73e5i)T + (-1.32e13 - 2.28e13i)T^{2} \) |
| 37 | \( 1 + (-1.55e6 - 2.70e6i)T + (-6.49e13 + 1.12e14i)T^{2} \) |
| 41 | \( 1 + 6.97e6iT - 3.27e14T^{2} \) |
| 43 | \( 1 + 1.88e7iT - 5.02e14T^{2} \) |
| 47 | \( 1 + (2.26e7 + 3.91e7i)T + (-5.59e14 + 9.69e14i)T^{2} \) |
| 53 | \( 1 + (-9.74e5 + 1.68e6i)T + (-1.64e15 - 2.85e15i)T^{2} \) |
| 59 | \( 1 + (-6.20e7 + 1.07e8i)T + (-4.33e15 - 7.50e15i)T^{2} \) |
| 61 | \( 1 + (-2.30e7 + 1.33e7i)T + (5.84e15 - 1.01e16i)T^{2} \) |
| 67 | \( 1 + (-7.12e7 - 4.11e7i)T + (1.36e16 + 2.35e16i)T^{2} \) |
| 71 | \( 1 - 7.73e7iT - 4.58e16T^{2} \) |
| 73 | \( 1 + (1.35e8 + 7.79e7i)T + (2.94e16 + 5.09e16i)T^{2} \) |
| 79 | \( 1 + (4.27e8 - 2.46e8i)T + (5.99e16 - 1.03e17i)T^{2} \) |
| 83 | \( 1 + 2.68e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + (-2.58e8 + 1.49e8i)T + (1.75e17 - 3.03e17i)T^{2} \) |
| 97 | \( 1 + 1.65e9iT - 7.60e17T^{2} \) |
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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
−11.36397883045246884790018310725, −10.14248971459706804268536007231, −9.089680709060560639608364735632, −8.436085029140388969962936611828, −6.78268918493198606462927462037, −5.60865444365917532357021423853, −4.84285084931249071032518241691, −2.43827145752812325510462746304, −2.00540559384190681382214708529, −0.20687509903621187103148895723,
1.71708321115142524039654561240, 2.85294490789652265257227706720, 4.20854709812433005944066831692, 5.80080911015719991403653261127, 6.67831839757830802951008985416, 7.904084988915697281007792930182, 9.576657273013371330201515970128, 10.33275017845199941125692200321, 10.57011497287974458389054616879, 12.81693994094780782901008792861
