L(s) = 1 | + (−0.5 + 0.866i)3-s + 2·5-s + (0.5 + 0.866i)7-s + (1 + 1.73i)9-s + (−0.5 + 0.866i)11-s + (−1 − 3.46i)13-s + (−1 + 1.73i)15-s + (−1.5 − 2.59i)17-s + (−3.5 − 6.06i)19-s − 0.999·21-s + (−0.5 + 0.866i)23-s − 25-s − 5·27-s + (−1.5 + 2.59i)29-s + 8·31-s + ⋯ |
L(s) = 1 | + (−0.288 + 0.499i)3-s + 0.894·5-s + (0.188 + 0.327i)7-s + (0.333 + 0.577i)9-s + (−0.150 + 0.261i)11-s + (−0.277 − 0.960i)13-s + (−0.258 + 0.447i)15-s + (−0.363 − 0.630i)17-s + (−0.802 − 1.39i)19-s − 0.218·21-s + (−0.104 + 0.180i)23-s − 0.200·25-s − 0.962·27-s + (−0.278 + 0.482i)29-s + 1.43·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.01646 + 0.279359i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.01646 + 0.279359i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 13 | \( 1 + (1 + 3.46i)T \) |
good | 3 | \( 1 + (0.5 - 0.866i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 - 2T + 5T^{2} \) |
| 7 | \( 1 + (-0.5 - 0.866i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3.5 + 6.06i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.5 - 0.866i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1.5 - 2.59i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (5.5 - 9.52i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (5.5 + 9.52i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 12T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + (-4.5 - 7.79i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.5 - 7.79i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.5 + 2.59i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.5 - 4.33i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 + (-0.5 + 0.866i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.5 - 0.866i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−13.67721496264820850969332970863, −13.06098010197014093996098963555, −11.66850121768190590489068512824, −10.52085077904631868882676399988, −9.816946494193511401443504654190, −8.574120347462755734274642843815, −7.09637732839689137909266446190, −5.60900411333539946589243347304, −4.67460463112019886015593311308, −2.43897965716384504950270600961,
1.83228794018555686871150659946, 4.14345910193158530011875188027, 5.92135947979335187446502644488, 6.71491554184052603124923928704, 8.167516996140399180117337750497, 9.525959149125548442170278174527, 10.44024030447475025031996535876, 11.76004934409619127987703453911, 12.68662080982883412099499869639, 13.67338800469130127364851304743