| L(s) = 1 | + (14.4 − 25.0i)2-s + (−95.3 + 165. i)3-s + (−162. − 281. i)4-s + 1.75e3·5-s + (2.75e3 + 4.77e3i)6-s + (2.20e3 + 3.81e3i)7-s + 5.41e3·8-s + (−8.35e3 − 1.44e4i)9-s + (2.53e4 − 4.39e4i)10-s + (5.87e3 − 1.01e4i)11-s + 6.19e4·12-s + (−2.15e4 + 1.00e5i)13-s + 1.27e5·14-s + (−1.67e5 + 2.89e5i)15-s + (1.61e5 − 2.79e5i)16-s + (1.96e5 + 3.40e5i)17-s + ⋯ |
| L(s) = 1 | + (0.639 − 1.10i)2-s + (−0.679 + 1.17i)3-s + (−0.317 − 0.549i)4-s + 1.25·5-s + (0.869 + 1.50i)6-s + (0.346 + 0.600i)7-s + 0.467·8-s + (−0.424 − 0.734i)9-s + (0.801 − 1.38i)10-s + (0.121 − 0.209i)11-s + 0.862·12-s + (−0.209 + 0.977i)13-s + 0.886·14-s + (−0.852 + 1.47i)15-s + (0.615 − 1.06i)16-s + (0.571 + 0.989i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0441i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0441i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(2.29801 + 0.0507951i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.29801 + 0.0507951i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (2.15e4 - 1.00e5i)T \) |
| good | 2 | \( 1 + (-14.4 + 25.0i)T + (-256 - 443. i)T^{2} \) |
| 3 | \( 1 + (95.3 - 165. i)T + (-9.84e3 - 1.70e4i)T^{2} \) |
| 5 | \( 1 - 1.75e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + (-2.20e3 - 3.81e3i)T + (-2.01e7 + 3.49e7i)T^{2} \) |
| 11 | \( 1 + (-5.87e3 + 1.01e4i)T + (-1.17e9 - 2.04e9i)T^{2} \) |
| 17 | \( 1 + (-1.96e5 - 3.40e5i)T + (-5.92e10 + 1.02e11i)T^{2} \) |
| 19 | \( 1 + (4.14e5 + 7.17e5i)T + (-1.61e11 + 2.79e11i)T^{2} \) |
| 23 | \( 1 + (2.04e5 - 3.54e5i)T + (-9.00e11 - 1.55e12i)T^{2} \) |
| 29 | \( 1 + (-3.19e6 + 5.53e6i)T + (-7.25e12 - 1.25e13i)T^{2} \) |
| 31 | \( 1 + 6.65e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + (-1.12e7 + 1.95e7i)T + (-6.49e13 - 1.12e14i)T^{2} \) |
| 41 | \( 1 + (7.10e6 - 1.23e7i)T + (-1.63e14 - 2.83e14i)T^{2} \) |
| 43 | \( 1 + (-1.50e6 - 2.60e6i)T + (-2.51e14 + 4.35e14i)T^{2} \) |
| 47 | \( 1 - 2.19e6T + 1.11e15T^{2} \) |
| 53 | \( 1 + 4.09e6T + 3.29e15T^{2} \) |
| 59 | \( 1 + (3.37e7 + 5.84e7i)T + (-4.33e15 + 7.50e15i)T^{2} \) |
| 61 | \( 1 + (-6.92e7 - 1.19e8i)T + (-5.84e15 + 1.01e16i)T^{2} \) |
| 67 | \( 1 + (1.18e8 - 2.05e8i)T + (-1.36e16 - 2.35e16i)T^{2} \) |
| 71 | \( 1 + (1.06e8 + 1.84e8i)T + (-2.29e16 + 3.97e16i)T^{2} \) |
| 73 | \( 1 + 3.17e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 3.54e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 2.35e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + (-6.81e7 + 1.18e8i)T + (-1.75e17 - 3.03e17i)T^{2} \) |
| 97 | \( 1 + (-4.83e7 - 8.37e7i)T + (-3.80e17 + 6.58e17i)T^{2} \) |
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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
−17.53962720622457187988597056105, −16.47283655062089172653807728703, −14.70104066155446087100951120065, −13.25463459674720155946723885246, −11.67785252774148232359133677665, −10.59064418195318369262889216692, −9.414608997387191566901282481224, −5.72603208378104932029183611492, −4.32027984369935042557039632555, −2.11102549853889575307970784685,
1.40729040174385411421076528372, 5.31323975554609459187163173245, 6.41255866609675852238765819016, 7.64171189704081400516132091858, 10.38212671464218973760174452967, 12.52958809427911377642808715226, 13.60805447970529965940093413182, 14.58938441555931511965426430129, 16.60330280868412478162600325003, 17.44633501400231767539147276304
