| L(s) = 1 | + (−2.39 − 0.448i)2-s + (0.0268 − 0.289i)3-s + (3.69 + 1.42i)4-s + (−1.02 + 1.35i)5-s + (−0.194 + 0.682i)6-s + (0.635 − 0.393i)7-s + (−4.06 − 2.51i)8-s + (2.86 + 0.535i)9-s + (3.06 − 2.79i)10-s + (3.11 + 0.582i)11-s + (0.512 − 1.03i)12-s + (4.09 − 2.53i)13-s + (−1.70 + 0.659i)14-s + (0.364 + 0.332i)15-s + (2.77 + 2.52i)16-s + (−0.126 − 0.445i)17-s + ⋯ |
| L(s) = 1 | + (−1.69 − 0.317i)2-s + (0.0154 − 0.167i)3-s + (1.84 + 0.714i)4-s + (−0.457 + 0.605i)5-s + (−0.0792 + 0.278i)6-s + (0.240 − 0.148i)7-s + (−1.43 − 0.889i)8-s + (0.955 + 0.178i)9-s + (0.968 − 0.882i)10-s + (0.940 + 0.175i)11-s + (0.148 − 0.297i)12-s + (1.13 − 0.702i)13-s + (−0.454 + 0.176i)14-s + (0.0941 + 0.0858i)15-s + (0.693 + 0.631i)16-s + (−0.0307 − 0.108i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0782i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0782i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.520985 - 0.0204150i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.520985 - 0.0204150i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 103 | \( 1 + (10.0 - 1.09i)T \) |
| good | 2 | \( 1 + (2.39 + 0.448i)T + (1.86 + 0.722i)T^{2} \) |
| 3 | \( 1 + (-0.0268 + 0.289i)T + (-2.94 - 0.551i)T^{2} \) |
| 5 | \( 1 + (1.02 - 1.35i)T + (-1.36 - 4.80i)T^{2} \) |
| 7 | \( 1 + (-0.635 + 0.393i)T + (3.12 - 6.26i)T^{2} \) |
| 11 | \( 1 + (-3.11 - 0.582i)T + (10.2 + 3.97i)T^{2} \) |
| 13 | \( 1 + (-4.09 + 2.53i)T + (5.79 - 11.6i)T^{2} \) |
| 17 | \( 1 + (0.126 + 0.445i)T + (-14.4 + 8.94i)T^{2} \) |
| 19 | \( 1 + (-0.484 - 5.22i)T + (-18.6 + 3.49i)T^{2} \) |
| 23 | \( 1 + (5.36 - 1.00i)T + (21.4 - 8.30i)T^{2} \) |
| 29 | \( 1 + (-3.03 + 4.01i)T + (-7.93 - 27.8i)T^{2} \) |
| 31 | \( 1 + (4.49 + 4.09i)T + (2.86 + 30.8i)T^{2} \) |
| 37 | \( 1 + (3.09 - 6.21i)T + (-22.2 - 29.5i)T^{2} \) |
| 41 | \( 1 + (-2.89 - 3.82i)T + (-11.2 + 39.4i)T^{2} \) |
| 43 | \( 1 + (4.30 + 8.64i)T + (-25.9 + 34.3i)T^{2} \) |
| 47 | \( 1 - 7.66T + 47T^{2} \) |
| 53 | \( 1 + (1.08 + 11.6i)T + (-52.0 + 9.73i)T^{2} \) |
| 59 | \( 1 + (9.12 + 5.65i)T + (26.2 + 52.8i)T^{2} \) |
| 61 | \( 1 + (0.668 + 2.35i)T + (-51.8 + 32.1i)T^{2} \) |
| 67 | \( 1 + (7.89 + 4.88i)T + (29.8 + 59.9i)T^{2} \) |
| 71 | \( 1 + (-0.692 - 0.916i)T + (-19.4 + 68.2i)T^{2} \) |
| 73 | \( 1 + (-3.45 - 4.57i)T + (-19.9 + 70.2i)T^{2} \) |
| 79 | \( 1 + (-6.49 + 8.60i)T + (-21.6 - 75.9i)T^{2} \) |
| 83 | \( 1 + (11.4 - 7.11i)T + (36.9 - 74.2i)T^{2} \) |
| 89 | \( 1 + (2.81 - 1.09i)T + (65.7 - 59.9i)T^{2} \) |
| 97 | \( 1 + (4.67 - 16.4i)T + (-82.4 - 51.0i)T^{2} \) |
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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
−13.79963982300557363669389748464, −12.28616566879001839984530990880, −11.34149172641491227041671891685, −10.44788468864307811688192464675, −9.618888774467653972313160031018, −8.254313666799499562755230798339, −7.54077244954644533232651262247, −6.39859431633816432691797477285, −3.71677318100479805561814887792, −1.57338093808768846769329436431,
1.36940829000815913323273070932, 4.24160741345773352212672777800, 6.36295319953672279481942703248, 7.38348024191834066061039334660, 8.749183130785576029012103160177, 9.075133101767736989235152455275, 10.42076552679592682915094001620, 11.37267713618231068237152812454, 12.44279052337179006443562715731, 14.02807035223509492948564262024
