L(s) = 1 | + (−1 − i)5-s + 4i·7-s + (4 + 4i)11-s + (3 − 3i)13-s − 6·17-s + (−4 + 4i)19-s + 8i·23-s − 3i·25-s + (−3 + 3i)29-s + 4·31-s + (4 − 4i)35-s + (1 + i)37-s − 2i·41-s + (−4 − 4i)43-s − 8·47-s + ⋯ |
L(s) = 1 | + (−0.447 − 0.447i)5-s + 1.51i·7-s + (1.20 + 1.20i)11-s + (0.832 − 0.832i)13-s − 1.45·17-s + (−0.917 + 0.917i)19-s + 1.66i·23-s − 0.600i·25-s + (−0.557 + 0.557i)29-s + 0.718·31-s + (0.676 − 0.676i)35-s + (0.164 + 0.164i)37-s − 0.312i·41-s + (−0.609 − 0.609i)43-s − 1.16·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 - 0.382i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9338492693\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9338492693\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (1 + i)T + 5iT^{2} \) |
| 7 | \( 1 - 4iT - 7T^{2} \) |
| 11 | \( 1 + (-4 - 4i)T + 11iT^{2} \) |
| 13 | \( 1 + (-3 + 3i)T - 13iT^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 + (4 - 4i)T - 19iT^{2} \) |
| 23 | \( 1 - 8iT - 23T^{2} \) |
| 29 | \( 1 + (3 - 3i)T - 29iT^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + (-1 - i)T + 37iT^{2} \) |
| 41 | \( 1 + 2iT - 41T^{2} \) |
| 43 | \( 1 + (4 + 4i)T + 43iT^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 + (-7 - 7i)T + 53iT^{2} \) |
| 59 | \( 1 + 59iT^{2} \) |
| 61 | \( 1 + (3 - 3i)T - 61iT^{2} \) |
| 67 | \( 1 + (-8 + 8i)T - 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + 10iT - 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 + (4 - 4i)T - 83iT^{2} \) |
| 89 | \( 1 + 16iT - 89T^{2} \) |
| 97 | \( 1 - 8T + 97T^{2} \) |
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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
−8.673696334053679521851080220089, −8.125252898525734162345738845524, −7.17167921341114506426061461828, −6.32919300812062991399649189362, −5.81489424590002904063473204570, −4.89162157931904500059247578908, −4.17228181895958762948547465419, −3.38504639685471169939439757536, −2.18079949349393398136829352827, −1.51507302033249518543380203513,
0.26498777618706456896921579746, 1.27527104327676576815016109090, 2.55160423864811998730976097456, 3.71541901665245164711556887685, 4.06616906829066569141482426350, 4.71147686354858537821919516533, 6.21489734747117650109572719742, 6.71375425939422245946807830270, 6.91273493175712294257869481862, 8.148589144842269016711639958496