| L(s) = 1 | + (1.28 − 0.948i)2-s + (−0.110 + 0.585i)3-s + (0.163 − 0.529i)4-s + (0.775 + 2.09i)5-s + (0.413 + 0.857i)6-s + (−0.606 − 2.57i)7-s + (0.763 + 2.18i)8-s + (2.46 + 0.966i)9-s + (2.98 + 1.96i)10-s + (1.01 + 2.59i)11-s + (0.291 + 0.154i)12-s + (−0.754 − 6.69i)13-s + (−3.22 − 2.73i)14-s + (−1.31 + 0.221i)15-s + (3.96 + 2.70i)16-s + (0.0996 + 2.66i)17-s + ⋯ |
| L(s) = 1 | + (0.909 − 0.670i)2-s + (−0.0639 + 0.337i)3-s + (0.0815 − 0.264i)4-s + (0.346 + 0.938i)5-s + (0.168 + 0.350i)6-s + (−0.229 − 0.973i)7-s + (0.269 + 0.771i)8-s + (0.820 + 0.322i)9-s + (0.944 + 0.620i)10-s + (0.306 + 0.781i)11-s + (0.0841 + 0.0444i)12-s + (−0.209 − 1.85i)13-s + (−0.861 − 0.731i)14-s + (−0.339 + 0.0571i)15-s + (0.991 + 0.676i)16-s + (0.0241 + 0.646i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0147i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0147i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.98846 - 0.0146268i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.98846 - 0.0146268i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-0.775 - 2.09i)T \) |
| 7 | \( 1 + (0.606 + 2.57i)T \) |
| good | 2 | \( 1 + (-1.28 + 0.948i)T + (0.589 - 1.91i)T^{2} \) |
| 3 | \( 1 + (0.110 - 0.585i)T + (-2.79 - 1.09i)T^{2} \) |
| 11 | \( 1 + (-1.01 - 2.59i)T + (-8.06 + 7.48i)T^{2} \) |
| 13 | \( 1 + (0.754 + 6.69i)T + (-12.6 + 2.89i)T^{2} \) |
| 17 | \( 1 + (-0.0996 - 2.66i)T + (-16.9 + 1.27i)T^{2} \) |
| 19 | \( 1 + (2.69 + 4.67i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (5.61 + 0.209i)T + (22.9 + 1.71i)T^{2} \) |
| 29 | \( 1 + (-1.09 - 0.250i)T + (26.1 + 12.5i)T^{2} \) |
| 31 | \( 1 + (3.55 + 2.05i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.58 + 6.77i)T + (-20.8 - 30.5i)T^{2} \) |
| 41 | \( 1 + (-2.43 + 5.05i)T + (-25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (0.871 + 0.305i)T + (33.6 + 26.8i)T^{2} \) |
| 47 | \( 1 + (2.24 + 3.03i)T + (-13.8 + 44.9i)T^{2} \) |
| 53 | \( 1 + (1.13 + 2.15i)T + (-29.8 + 43.7i)T^{2} \) |
| 59 | \( 1 + (0.773 - 10.3i)T + (-58.3 - 8.79i)T^{2} \) |
| 61 | \( 1 + (-1.50 - 4.87i)T + (-50.4 + 34.3i)T^{2} \) |
| 67 | \( 1 + (-1.34 - 5.02i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (1.07 + 4.69i)T + (-63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (-3.24 + 4.39i)T + (-21.5 - 69.7i)T^{2} \) |
| 79 | \( 1 + (-6.60 + 3.81i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-6.68 - 0.753i)T + (80.9 + 18.4i)T^{2} \) |
| 89 | \( 1 + (5.20 - 13.2i)T + (-65.2 - 60.5i)T^{2} \) |
| 97 | \( 1 + (-10.8 - 10.8i)T + 97iT^{2} \) |
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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
−12.31781673700619408660294538292, −10.83721997320643734782011140586, −10.58696156392334153448226801573, −9.730071635851396235477487526590, −7.906373487260398673406685440027, −7.09306004538581537267319022791, −5.69490113556248880430891619027, −4.40503092888110369780249665935, −3.58513328501143147551223421388, −2.23824521871773766042525483603,
1.68822889261074839090081177202, 3.94959854242719829413443614793, 4.89517559353117475910305265698, 6.11588095575865266634442678143, 6.55096058259376466691821237705, 8.053818521307209616771984978175, 9.270715957091158208931866060566, 9.820894941254572031875558182075, 11.68074665978644855200205033661, 12.33372838099351800362149885749
