| L(s) = 1 | + (−1 + i)2-s − 2i·4-s + (2 + 2i)8-s + (1 + i)13-s − 4·16-s + (3 − 3i)17-s − 2·26-s + 4i·29-s + (4 − 4i)32-s + 6i·34-s + (7 − 7i)37-s + 8·41-s − 7i·49-s + (2 − 2i)52-s + (9 + 9i)53-s + ⋯ |
| L(s) = 1 | + (−0.707 + 0.707i)2-s − i·4-s + (0.707 + 0.707i)8-s + (0.277 + 0.277i)13-s − 16-s + (0.727 − 0.727i)17-s − 0.392·26-s + 0.742i·29-s + (0.707 − 0.707i)32-s + 1.02i·34-s + (1.15 − 1.15i)37-s + 1.24·41-s − i·49-s + (0.277 − 0.277i)52-s + (1.23 + 1.23i)53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.06512 + 0.302581i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.06512 + 0.302581i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1 - i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 7iT^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + (-1 - i)T + 13iT^{2} \) |
| 17 | \( 1 + (-3 + 3i)T - 17iT^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 23iT^{2} \) |
| 29 | \( 1 - 4iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + (-7 + 7i)T - 37iT^{2} \) |
| 41 | \( 1 - 8T + 41T^{2} \) |
| 43 | \( 1 - 43iT^{2} \) |
| 47 | \( 1 + 47iT^{2} \) |
| 53 | \( 1 + (-9 - 9i)T + 53iT^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 12T + 61T^{2} \) |
| 67 | \( 1 + 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + (-11 - 11i)T + 73iT^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 83iT^{2} \) |
| 89 | \( 1 + 16iT - 89T^{2} \) |
| 97 | \( 1 + (13 - 13i)T - 97iT^{2} \) |
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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
−9.958326190281891474273146774728, −9.284883462273930079427006954417, −8.518783887235664801314418914145, −7.59712923891881673015669410983, −6.95771576889232062784850323731, −5.92918359433554534305807609298, −5.18908606733813630322571220897, −4.01029074418267914517628519757, −2.44730792546244114735843260550, −0.945154214135935140126786809495,
0.992525463155269707789615012909, 2.35899830498500443573934550327, 3.46835915504507015116038036725, 4.39892952677900730560774447126, 5.74112646552689243256206578988, 6.77522288810702706279349393391, 7.87422519838173383075382223527, 8.308463138288762191516742471610, 9.393580817275264422290883811800, 9.996890277123390682354009991054
