L(s) = 1 | + (1 + i)3-s + (2 − 2i)5-s + 4i·7-s − i·9-s + (1 − i)11-s + (−2 − 2i)13-s + 4·15-s + 4·17-s + (5 + 5i)19-s + (−4 + 4i)21-s − 4i·23-s − 3i·25-s + (4 − 4i)27-s + (6 + 6i)29-s − 8·31-s + ⋯ |
L(s) = 1 | + (0.577 + 0.577i)3-s + (0.894 − 0.894i)5-s + 1.51i·7-s − 0.333i·9-s + (0.301 − 0.301i)11-s + (−0.554 − 0.554i)13-s + 1.03·15-s + 0.970·17-s + (1.14 + 1.14i)19-s + (−0.872 + 0.872i)21-s − 0.834i·23-s − 0.600i·25-s + (0.769 − 0.769i)27-s + (1.11 + 1.11i)29-s − 1.43·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.374732584\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.374732584\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 + (-1 - i)T + 3iT^{2} \) |
| 5 | \( 1 + (-2 + 2i)T - 5iT^{2} \) |
| 7 | \( 1 - 4iT - 7T^{2} \) |
| 11 | \( 1 + (-1 + i)T - 11iT^{2} \) |
| 13 | \( 1 + (2 + 2i)T + 13iT^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 + (-5 - 5i)T + 19iT^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 + (-6 - 6i)T + 29iT^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + (2 - 2i)T - 37iT^{2} \) |
| 41 | \( 1 - 2iT - 41T^{2} \) |
| 43 | \( 1 + (-3 + 3i)T - 43iT^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + (-2 + 2i)T - 53iT^{2} \) |
| 59 | \( 1 + (3 - 3i)T - 59iT^{2} \) |
| 61 | \( 1 + (6 + 6i)T + 61iT^{2} \) |
| 67 | \( 1 + (-3 - 3i)T + 67iT^{2} \) |
| 71 | \( 1 + 4iT - 71T^{2} \) |
| 73 | \( 1 + 4iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + (-7 - 7i)T + 83iT^{2} \) |
| 89 | \( 1 + 12iT - 89T^{2} \) |
| 97 | \( 1 + 4T + 97T^{2} \) |
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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
−9.764615062572975740680316170028, −9.114660113549770594600217856637, −8.705849885589958955331077862596, −7.79368861092914191011954337892, −6.32642198332482874502793387980, −5.50051297921217524724396112020, −5.04754153100810527987393446089, −3.55320615002066118266616457283, −2.70611621944822741206070694626, −1.38050411544664894275956908000,
1.28232496085313326742765465011, 2.41235962402359011867809460134, 3.37334240331858666483117869211, 4.57552259330940661498552565175, 5.70331115426585224516004049735, 7.00205120601783704123358472604, 7.15776588247263638959807987753, 7.927297533229742347163524709624, 9.301512972911754736804095257523, 9.861267452902821142165346095531