| L(s) = 1 | + (9.79 − 16.9i)2-s + (104. + 181. i)3-s + (63.9 + 110. i)4-s + (983. − 1.70e3i)5-s + 4.11e3·6-s + (−5.76e3 + 2.66e3i)7-s + 1.25e4·8-s + (−1.21e4 + 2.11e4i)9-s + (−1.92e4 − 3.33e4i)10-s + (−1.58e4 − 2.75e4i)11-s + (−1.34e4 + 2.32e4i)12-s − 1.00e5·13-s + (−1.13e4 + 1.23e5i)14-s + 4.13e5·15-s + (9.01e4 − 1.56e5i)16-s + (−1.69e5 − 2.92e5i)17-s + ⋯ |
| L(s) = 1 | + (0.433 − 0.750i)2-s + (0.748 + 1.29i)3-s + (0.124 + 0.216i)4-s + (0.703 − 1.21i)5-s + 1.29·6-s + (−0.908 + 0.418i)7-s + 1.08·8-s + (−0.619 + 1.07i)9-s + (−0.609 − 1.05i)10-s + (−0.327 − 0.567i)11-s + (−0.187 + 0.323i)12-s − 0.972·13-s + (−0.0791 + 0.862i)14-s + 2.10·15-s + (0.343 − 0.595i)16-s + (−0.491 − 0.850i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0280i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0280i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(2.26682 - 0.0318501i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.26682 - 0.0318501i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (5.76e3 - 2.66e3i)T \) |
| good | 2 | \( 1 + (-9.79 + 16.9i)T + (-256 - 443. i)T^{2} \) |
| 3 | \( 1 + (-104. - 181. i)T + (-9.84e3 + 1.70e4i)T^{2} \) |
| 5 | \( 1 + (-983. + 1.70e3i)T + (-9.76e5 - 1.69e6i)T^{2} \) |
| 11 | \( 1 + (1.58e4 + 2.75e4i)T + (-1.17e9 + 2.04e9i)T^{2} \) |
| 13 | \( 1 + 1.00e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + (1.69e5 + 2.92e5i)T + (-5.92e10 + 1.02e11i)T^{2} \) |
| 19 | \( 1 + (1.30e5 - 2.26e5i)T + (-1.61e11 - 2.79e11i)T^{2} \) |
| 23 | \( 1 + (-2.72e5 + 4.71e5i)T + (-9.00e11 - 1.55e12i)T^{2} \) |
| 29 | \( 1 - 2.32e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + (-2.53e6 - 4.39e6i)T + (-1.32e13 + 2.28e13i)T^{2} \) |
| 37 | \( 1 + (2.93e6 - 5.07e6i)T + (-6.49e13 - 1.12e14i)T^{2} \) |
| 41 | \( 1 + 2.81e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 3.88e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + (1.26e7 - 2.19e7i)T + (-5.59e14 - 9.69e14i)T^{2} \) |
| 53 | \( 1 + (-2.26e7 - 3.92e7i)T + (-1.64e15 + 2.85e15i)T^{2} \) |
| 59 | \( 1 + (9.44e6 + 1.63e7i)T + (-4.33e15 + 7.50e15i)T^{2} \) |
| 61 | \( 1 + (5.08e7 - 8.80e7i)T + (-5.84e15 - 1.01e16i)T^{2} \) |
| 67 | \( 1 + (6.55e7 + 1.13e8i)T + (-1.36e16 + 2.35e16i)T^{2} \) |
| 71 | \( 1 - 9.51e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + (1.19e8 + 2.06e8i)T + (-2.94e16 + 5.09e16i)T^{2} \) |
| 79 | \( 1 + (9.42e7 - 1.63e8i)T + (-5.99e16 - 1.03e17i)T^{2} \) |
| 83 | \( 1 + 1.76e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + (-2.41e8 + 4.18e8i)T + (-1.75e17 - 3.03e17i)T^{2} \) |
| 97 | \( 1 + 2.28e7T + 7.60e17T^{2} \) |
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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
−20.64810076760738362131648991702, −19.61299831093284778591446200969, −16.77980696676045805751893632960, −15.83700369985425614949608833490, −13.74801547607853015698033933295, −12.40115255293255044915839817728, −10.14214030117615246687642404925, −8.870447099971057783281853159394, −4.77405787646261198101472917143, −2.84092081527992741803542361822,
2.31640850783122798197191403210, 6.46103341581772464492173359665, 7.30600903137144338908787238906, 10.21585976865035642882221452280, 13.03985442963694668424679933821, 14.05863722979603759735263763744, 15.17349139297539854388547486858, 17.43207404657944917616647807534, 18.97494582938217957225889993626, 19.79794705052046195693179636307
