L(s) = 1 | + (0.396 − 2.50i)2-s + (−1.78 − 0.910i)3-s + (−4.21 − 1.36i)4-s + (−2.22 + 0.188i)5-s + (−2.98 + 4.11i)6-s + (0.820 + 1.61i)7-s + (−2.79 + 5.48i)8-s + (0.599 + 0.824i)9-s + (−0.412 + 5.65i)10-s + (6.27 + 6.27i)12-s + (−2.22 − 0.352i)13-s + (4.35 − 1.41i)14-s + (4.15 + 1.69i)15-s + (5.46 + 3.96i)16-s + (3.32 − 0.526i)17-s + (2.30 − 1.17i)18-s + ⋯ |
L(s) = 1 | + (0.280 − 1.77i)2-s + (−1.03 − 0.525i)3-s + (−2.10 − 0.684i)4-s + (−0.996 + 0.0840i)5-s + (−1.21 + 1.67i)6-s + (0.310 + 0.608i)7-s + (−0.987 + 1.93i)8-s + (0.199 + 0.274i)9-s + (−0.130 + 1.78i)10-s + (1.81 + 1.81i)12-s + (−0.617 − 0.0978i)13-s + (1.16 − 0.378i)14-s + (1.07 + 0.436i)15-s + (1.36 + 0.991i)16-s + (0.805 − 0.127i)17-s + (0.542 − 0.276i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 + 0.298i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.954 + 0.298i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.252952 - 0.0386081i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.252952 - 0.0386081i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (2.22 - 0.188i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.396 + 2.50i)T + (-1.90 - 0.618i)T^{2} \) |
| 3 | \( 1 + (1.78 + 0.910i)T + (1.76 + 2.42i)T^{2} \) |
| 7 | \( 1 + (-0.820 - 1.61i)T + (-4.11 + 5.66i)T^{2} \) |
| 13 | \( 1 + (2.22 + 0.352i)T + (12.3 + 4.01i)T^{2} \) |
| 17 | \( 1 + (-3.32 + 0.526i)T + (16.1 - 5.25i)T^{2} \) |
| 19 | \( 1 + (0.403 + 1.24i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (3.16 - 3.16i)T - 23iT^{2} \) |
| 29 | \( 1 + (0.907 - 2.79i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-3.24 + 2.35i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (2.85 - 1.45i)T + (21.7 - 29.9i)T^{2} \) |
| 41 | \( 1 + (1.23 - 0.402i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (4.55 + 4.55i)T + 43iT^{2} \) |
| 47 | \( 1 + (3.50 - 6.88i)T + (-27.6 - 38.0i)T^{2} \) |
| 53 | \( 1 + (0.996 - 6.29i)T + (-50.4 - 16.3i)T^{2} \) |
| 59 | \( 1 + (-11.7 - 3.82i)T + (47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (5.47 - 7.53i)T + (-18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (2.46 + 2.46i)T + 67iT^{2} \) |
| 71 | \( 1 + (-5.47 - 3.98i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (8.41 - 4.28i)T + (42.9 - 59.0i)T^{2} \) |
| 79 | \( 1 + (-0.926 + 0.673i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-1.63 - 10.3i)T + (-78.9 + 25.6i)T^{2} \) |
| 89 | \( 1 - 3.85iT - 89T^{2} \) |
| 97 | \( 1 + (-0.768 - 0.121i)T + (92.2 + 29.9i)T^{2} \) |
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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
−10.95127717584198950573140307210, −10.16745825438931438555779001158, −9.155382911611849993077596562781, −8.140281646387641383600663686656, −7.02372765905960978801642407167, −5.62663859448035394887258539600, −4.90420116486672450376934732002, −3.77180441868517194422215943704, −2.68547447462953514083907365233, −1.21792764829839554741926918618,
0.17185663139096250433403578480, 3.72954512570845808227260210672, 4.60003612955972789521145157845, 5.15514711176907737098581288534, 6.18460435704190212450476948824, 7.03529762021410508023348449811, 7.897338344821671745836109132569, 8.424739326509791396085189914414, 9.794956845778261069640435630364, 10.60658208407431752757795474542