| L(s) = 1 | − 3-s − 4i·5-s + 9-s − 2i·11-s + 4i·15-s − 2·17-s − 8i·19-s − 4·23-s − 11·25-s − 27-s − 6·29-s + 4i·31-s + 2i·33-s + 6i·37-s + 12i·41-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.78i·5-s + 0.333·9-s − 0.603i·11-s + 1.03i·15-s − 0.485·17-s − 1.83i·19-s − 0.834·23-s − 2.20·25-s − 0.192·27-s − 1.11·29-s + 0.718i·31-s + 0.348i·33-s + 0.986i·37-s + 1.87i·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.554 - 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.554 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3722574923\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3722574923\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 4iT - 5T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 2iT - 11T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 8iT - 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 4iT - 31T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 - 12iT - 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + 6iT - 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 + 14iT - 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 + 2iT - 71T^{2} \) |
| 73 | \( 1 + 2iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 14iT - 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 10iT - 97T^{2} \) |
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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
−8.226660941077146572274547627181, −7.18914116641740289640708927112, −6.37311257314733053783950046564, −5.58206465061934951192235524288, −4.88930915865677718139200468288, −4.48536748280519740277606260594, −3.42141294039053688788336290104, −2.05941343744698439545371308178, −1.02532474988957540364259615939, −0.12714425726434420776516908787,
1.82773280301130631086205581584, 2.49689490673139884913913650682, 3.77910002239430858907043900061, 4.04026961310307234749478073424, 5.56173143830816768412169856756, 5.93812720205782917887906178064, 6.74724172232196270335050684510, 7.39494316028735243940451077596, 7.82437074055018300681173828612, 8.997288635621098660284911847112
