L(s) = 1 | + (−0.984 + 0.173i)2-s + (−1.37 + 1.05i)3-s + (0.939 − 0.342i)4-s + (1.98 + 1.02i)5-s + (1.16 − 1.27i)6-s + (3.69 + 2.13i)7-s + (−0.866 + 0.5i)8-s + (0.770 − 2.89i)9-s + (−2.13 − 0.665i)10-s + (2.36 − 1.36i)11-s + (−0.929 + 1.46i)12-s + (2.74 + 2.30i)13-s + (−4.01 − 1.46i)14-s + (−3.81 + 0.689i)15-s + (0.766 − 0.642i)16-s + (−1.27 − 7.23i)17-s + ⋯ |
L(s) = 1 | + (−0.696 + 0.122i)2-s + (−0.792 + 0.609i)3-s + (0.469 − 0.171i)4-s + (0.888 + 0.458i)5-s + (0.477 − 0.521i)6-s + (1.39 + 0.806i)7-s + (−0.306 + 0.176i)8-s + (0.256 − 0.966i)9-s + (−0.675 − 0.210i)10-s + (0.712 − 0.411i)11-s + (−0.268 + 0.421i)12-s + (0.762 + 0.639i)13-s + (−1.07 − 0.390i)14-s + (−0.984 + 0.178i)15-s + (0.191 − 0.160i)16-s + (−0.309 − 1.75i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.615 - 0.787i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.615 - 0.787i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.07811 + 0.525797i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.07811 + 0.525797i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.984 - 0.173i)T \) |
| 3 | \( 1 + (1.37 - 1.05i)T \) |
| 5 | \( 1 + (-1.98 - 1.02i)T \) |
| 19 | \( 1 + (-0.726 + 4.29i)T \) |
good | 7 | \( 1 + (-3.69 - 2.13i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.36 + 1.36i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.74 - 2.30i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (1.27 + 7.23i)T + (-15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (0.504 - 0.183i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (-1.17 + 6.65i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-6.25 - 3.61i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 7.30T + 37T^{2} \) |
| 41 | \( 1 + (4.84 - 4.06i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-0.172 + 0.473i)T + (-32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (0.399 - 2.26i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (-0.105 - 0.291i)T + (-40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (-1.70 - 9.66i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-1.45 + 0.530i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (0.976 - 5.53i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-2.79 - 1.01i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (8.13 + 9.69i)T + (-12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (9.01 + 10.7i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (7.26 - 12.5i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (7.27 + 6.10i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (-0.308 - 1.75i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−10.92305787708217153968510988113, −9.943214249701130274346730104654, −9.109681501953178089247417102804, −8.612333885101119824562698531228, −7.08767440525249639412806145727, −6.33485110312094902736685849153, −5.41637194719334237848000840619, −4.57634961806892719259148124398, −2.76151029553588992173255723072, −1.33392103089017093632656996184,
1.31578907532047751515786608129, 1.73661430586358951288052884288, 4.04577396715638868204858968420, 5.23285065374675465176964665909, 6.14722151399694390816803502964, 7.01490860258442343205404515608, 8.194810262899959449903106117865, 8.515052434146010185254697930255, 10.11009225598467079937429041086, 10.51281807944864759615593498789