| L(s) = 1 | + (−29.6 + 17.1i)2-s + (−132. − 228. i)3-s + (330. − 571. i)4-s − 402. i·5-s + (7.82e3 + 4.51e3i)6-s + (−8.11e3 − 4.68e3i)7-s + 5.06e3i·8-s + (−2.50e4 + 4.33e4i)9-s + (6.88e3 + 1.19e4i)10-s + (1.54e4 − 8.90e3i)11-s − 1.74e5·12-s + (7.06e4 + 7.49e4i)13-s + 3.20e5·14-s + (−9.19e4 + 5.31e4i)15-s + (8.22e4 + 1.42e5i)16-s + (2.32e4 − 4.03e4i)17-s + ⋯ |
| L(s) = 1 | + (−1.31 + 0.756i)2-s + (−0.940 − 1.62i)3-s + (0.644 − 1.11i)4-s − 0.287i·5-s + (2.46 + 1.42i)6-s + (−1.27 − 0.737i)7-s + 0.437i·8-s + (−1.27 + 2.20i)9-s + (0.217 + 0.377i)10-s + (0.317 − 0.183i)11-s − 2.42·12-s + (0.686 + 0.727i)13-s + 2.23·14-s + (−0.469 + 0.270i)15-s + (0.313 + 0.543i)16-s + (0.0675 − 0.117i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.218 - 0.975i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.218 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.0359908 + 0.0449199i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0359908 + 0.0449199i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (-7.06e4 - 7.49e4i)T \) |
| good | 2 | \( 1 + (29.6 - 17.1i)T + (256 - 443. i)T^{2} \) |
| 3 | \( 1 + (132. + 228. i)T + (-9.84e3 + 1.70e4i)T^{2} \) |
| 5 | \( 1 + 402. iT - 1.95e6T^{2} \) |
| 7 | \( 1 + (8.11e3 + 4.68e3i)T + (2.01e7 + 3.49e7i)T^{2} \) |
| 11 | \( 1 + (-1.54e4 + 8.90e3i)T + (1.17e9 - 2.04e9i)T^{2} \) |
| 17 | \( 1 + (-2.32e4 + 4.03e4i)T + (-5.92e10 - 1.02e11i)T^{2} \) |
| 19 | \( 1 + (2.51e5 + 1.45e5i)T + (1.61e11 + 2.79e11i)T^{2} \) |
| 23 | \( 1 + (8.16e5 + 1.41e6i)T + (-9.00e11 + 1.55e12i)T^{2} \) |
| 29 | \( 1 + (1.75e5 + 3.04e5i)T + (-7.25e12 + 1.25e13i)T^{2} \) |
| 31 | \( 1 - 6.33e6iT - 2.64e13T^{2} \) |
| 37 | \( 1 + (1.06e7 - 6.13e6i)T + (6.49e13 - 1.12e14i)T^{2} \) |
| 41 | \( 1 + (2.52e6 - 1.45e6i)T + (1.63e14 - 2.83e14i)T^{2} \) |
| 43 | \( 1 + (2.83e6 - 4.90e6i)T + (-2.51e14 - 4.35e14i)T^{2} \) |
| 47 | \( 1 + 1.35e7iT - 1.11e15T^{2} \) |
| 53 | \( 1 - 5.34e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + (8.14e7 + 4.70e7i)T + (4.33e15 + 7.50e15i)T^{2} \) |
| 61 | \( 1 + (7.82e7 - 1.35e8i)T + (-5.84e15 - 1.01e16i)T^{2} \) |
| 67 | \( 1 + (1.50e8 - 8.71e7i)T + (1.36e16 - 2.35e16i)T^{2} \) |
| 71 | \( 1 + (9.79e6 + 5.65e6i)T + (2.29e16 + 3.97e16i)T^{2} \) |
| 73 | \( 1 + 8.79e7iT - 5.88e16T^{2} \) |
| 79 | \( 1 - 3.00e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 7.05e8iT - 1.86e17T^{2} \) |
| 89 | \( 1 + (4.64e8 - 2.68e8i)T + (1.75e17 - 3.03e17i)T^{2} \) |
| 97 | \( 1 + (-1.94e8 - 1.12e8i)T + (3.80e17 + 6.58e17i)T^{2} \) |
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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
−18.03865138829707888978788380080, −16.76792020939973398655779558065, −16.37625818387532477987134223569, −13.61727289263345745862411726835, −12.40409790227170641348701802264, −10.58828017876137512725163920891, −8.620191658711198375596086383097, −6.97849885225450933537466177701, −6.36686800313681186165959894170, −1.12466446186515263852269366091,
0.06532031699282949588703967749, 3.35588321488004131645965664097, 5.88947573333368248204089138617, 9.007971184333308046999869904463, 9.934382566076870518194725668338, 10.88817123122628854886415716231, 12.16984685162834603591955314793, 15.23648408485991098899234634876, 16.26239877014758169029201210049, 17.32531430225017745395516626199
