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
−9.861267452902821142165346095531, −9.301512972911754736804095257523, −7.927297533229742347163524709624, −7.15776588247263638959807987753, −7.00205120601783704123358472604, −5.70331115426585224516004049735, −4.57552259330940661498552565175, −3.37334240331858666483117869211, −2.41235962402359011867809460134, −1.28232496085313326742765465011,
1.38050411544664894275956908000, 2.70611621944822741206070694626, 3.55320615002066118266616457283, 5.04754153100810527987393446089, 5.50051297921217524724396112020, 6.32642198332482874502793387980, 7.79368861092914191011954337892, 8.705849885589958955331077862596, 9.114660113549770594600217856637, 9.764615062572975740680316170028