L(s) = 1 | + (−0.418 − 1.28i)2-s + (−0.465 − 0.338i)3-s + (0.131 − 0.0953i)4-s + (−0.309 + 0.951i)5-s + (−0.241 + 0.741i)6-s + (−2.95 + 2.14i)7-s + (−2.37 − 1.72i)8-s + (−0.824 − 2.53i)9-s + 1.35·10-s − 0.0933·12-s + (0.874 + 2.69i)13-s + (4.00 + 2.90i)14-s + (0.465 − 0.338i)15-s + (−1.12 + 3.47i)16-s + (−1.14 + 3.51i)17-s + (−2.92 + 2.12i)18-s + ⋯ |
L(s) = 1 | + (−0.296 − 0.911i)2-s + (−0.268 − 0.195i)3-s + (0.0655 − 0.0476i)4-s + (−0.138 + 0.425i)5-s + (−0.0984 + 0.302i)6-s + (−1.11 + 0.810i)7-s + (−0.838 − 0.609i)8-s + (−0.274 − 0.846i)9-s + 0.428·10-s − 0.0269·12-s + (0.242 + 0.746i)13-s + (1.06 + 0.777i)14-s + (0.120 − 0.0873i)15-s + (−0.281 + 0.867i)16-s + (−0.276 + 0.851i)17-s + (−0.689 + 0.501i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.263 - 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.263 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.244714 + 0.186842i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.244714 + 0.186842i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (0.309 - 0.951i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.418 + 1.28i)T + (-1.61 + 1.17i)T^{2} \) |
| 3 | \( 1 + (0.465 + 0.338i)T + (0.927 + 2.85i)T^{2} \) |
| 7 | \( 1 + (2.95 - 2.14i)T + (2.16 - 6.65i)T^{2} \) |
| 13 | \( 1 + (-0.874 - 2.69i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (1.14 - 3.51i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (0.0769 + 0.0559i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 - 1.16T + 23T^{2} \) |
| 29 | \( 1 + (5.46 - 3.96i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-2.09 - 6.44i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (7.96 - 5.78i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (6.72 + 4.88i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - 2.96T + 43T^{2} \) |
| 47 | \( 1 + (-1.79 - 1.30i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (-0.925 - 2.84i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-6.88 + 5.00i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-2.62 + 8.06i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + 13.4T + 67T^{2} \) |
| 71 | \( 1 + (-2.56 + 7.89i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (1.06 - 0.775i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (-4.28 - 13.1i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-3.28 + 10.1i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + 12.1T + 89T^{2} \) |
| 97 | \( 1 + (1.33 + 4.11i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−10.86335561627544301458660749941, −10.08483901810932481625504072780, −9.182821583633509457498070929211, −8.699818262700436626421100411724, −6.84688255174900889381598707906, −6.51521294034775634799761656435, −5.57224411186557502739796976858, −3.68644025371325613403267508495, −3.02290619402642691214279821999, −1.69736431063964018481726433243,
0.18195150161466740869011237138, 2.63812723761394884307158217394, 3.88307999277936950770045276328, 5.23062667042090140249421419411, 5.99073242904696390292938082022, 7.03902264317442229100661232895, 7.65900162315775146964799784256, 8.553279824332233970262953708370, 9.500653728825121104335006172045, 10.37268523434410534051835139786