| L(s) = 1 | + (17.2 + 29.9i)2-s + (−28.8 − 50.0i)3-s + (−340. + 590. i)4-s − 2.01e3·5-s + (998. − 1.72e3i)6-s + (−5.87e3 + 1.01e4i)7-s − 5.86e3·8-s + (8.17e3 − 1.41e4i)9-s + (−3.48e4 − 6.03e4i)10-s + (−1.01e3 − 1.76e3i)11-s + 3.94e4·12-s + (4.33e4 + 9.33e4i)13-s − 4.06e5·14-s + (5.82e4 + 1.00e5i)15-s + (7.32e4 + 1.26e5i)16-s + (8.24e4 − 1.42e5i)17-s + ⋯ |
| L(s) = 1 | + (0.763 + 1.32i)2-s + (−0.205 − 0.356i)3-s + (−0.665 + 1.15i)4-s − 1.44·5-s + (0.314 − 0.544i)6-s + (−0.925 + 1.60i)7-s − 0.506·8-s + (0.415 − 0.719i)9-s + (−1.10 − 1.90i)10-s + (−0.0209 − 0.0363i)11-s + 0.548·12-s + (0.421 + 0.906i)13-s − 2.82·14-s + (0.297 + 0.514i)15-s + (0.279 + 0.483i)16-s + (0.239 − 0.414i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 + 0.178i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.983 + 0.178i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.105224 - 1.16934i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.105224 - 1.16934i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (-4.33e4 - 9.33e4i)T \) |
| good | 2 | \( 1 + (-17.2 - 29.9i)T + (-256 + 443. i)T^{2} \) |
| 3 | \( 1 + (28.8 + 50.0i)T + (-9.84e3 + 1.70e4i)T^{2} \) |
| 5 | \( 1 + 2.01e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + (5.87e3 - 1.01e4i)T + (-2.01e7 - 3.49e7i)T^{2} \) |
| 11 | \( 1 + (1.01e3 + 1.76e3i)T + (-1.17e9 + 2.04e9i)T^{2} \) |
| 17 | \( 1 + (-8.24e4 + 1.42e5i)T + (-5.92e10 - 1.02e11i)T^{2} \) |
| 19 | \( 1 + (2.79e5 - 4.83e5i)T + (-1.61e11 - 2.79e11i)T^{2} \) |
| 23 | \( 1 + (-1.43e5 - 2.47e5i)T + (-9.00e11 + 1.55e12i)T^{2} \) |
| 29 | \( 1 + (-8.68e5 - 1.50e6i)T + (-7.25e12 + 1.25e13i)T^{2} \) |
| 31 | \( 1 + 3.33e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + (8.01e6 + 1.38e7i)T + (-6.49e13 + 1.12e14i)T^{2} \) |
| 41 | \( 1 + (-1.41e7 - 2.45e7i)T + (-1.63e14 + 2.83e14i)T^{2} \) |
| 43 | \( 1 + (-8.28e6 + 1.43e7i)T + (-2.51e14 - 4.35e14i)T^{2} \) |
| 47 | \( 1 - 2.16e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 7.19e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + (9.09e6 - 1.57e7i)T + (-4.33e15 - 7.50e15i)T^{2} \) |
| 61 | \( 1 + (4.77e7 - 8.26e7i)T + (-5.84e15 - 1.01e16i)T^{2} \) |
| 67 | \( 1 + (-4.24e7 - 7.35e7i)T + (-1.36e16 + 2.35e16i)T^{2} \) |
| 71 | \( 1 + (1.81e8 - 3.14e8i)T + (-2.29e16 - 3.97e16i)T^{2} \) |
| 73 | \( 1 + 3.37e7T + 5.88e16T^{2} \) |
| 79 | \( 1 - 3.19e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 1.88e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + (1.25e8 + 2.18e8i)T + (-1.75e17 + 3.03e17i)T^{2} \) |
| 97 | \( 1 + (-1.09e8 + 1.89e8i)T + (-3.80e17 - 6.58e17i)T^{2} \) |
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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
−18.40607736462783299370092318719, −16.32074616613782556107507447513, −15.68465266865936892706429342162, −14.72216058738154401801677974127, −12.74504250224481612010618142347, −11.90463374597248496715559773252, −8.819706037652899440031363700291, −7.17329954957905039280549870691, −5.91337156382688946634780134577, −3.84261937909313358093358052868,
0.53591568149049202396330997076, 3.47548163327196274441215145899, 4.42406088788290959470680755132, 7.51109727405791517566270188719, 10.34570494351739499209040407405, 11.00008755541562244834374140989, 12.61566147600699972271215913988, 13.59512929550235956995472838319, 15.54917196888817807891689500431, 16.74313286401022898097045398022
