| L(s) = 1 | + 2·2-s − 3-s + 2·4-s + (2 + i)5-s − 2·6-s + 4i·7-s + 9-s + (4 + 2i)10-s − i·11-s − 2·12-s + 2i·13-s + 8i·14-s + (−2 − i)15-s − 4·16-s + 6·17-s + 2·18-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.577·3-s + 4-s + (0.894 + 0.447i)5-s − 0.816·6-s + 1.51i·7-s + 0.333·9-s + (1.26 + 0.632i)10-s − 0.301i·11-s − 0.577·12-s + 0.554i·13-s + 2.13i·14-s + (−0.516 − 0.258i)15-s − 16-s + 1.45·17-s + 0.471·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.747 - 0.664i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.747 - 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.45992 + 0.935260i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.45992 + 0.935260i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 + (-2 - i)T \) |
| 29 | \( 1 + (2 - 5i)T \) |
| good | 2 | \( 1 - 2T + 2T^{2} \) |
| 7 | \( 1 - 4iT - 7T^{2} \) |
| 11 | \( 1 + iT - 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 + 9iT - 23T^{2} \) |
| 31 | \( 1 + 2iT - 31T^{2} \) |
| 37 | \( 1 - T + 37T^{2} \) |
| 41 | \( 1 + 9iT - 41T^{2} \) |
| 43 | \( 1 - T + 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 - 9iT - 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 + 6iT - 61T^{2} \) |
| 67 | \( 1 + 12iT - 67T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 + 15T + 73T^{2} \) |
| 79 | \( 1 - 4iT - 79T^{2} \) |
| 83 | \( 1 + 7iT - 83T^{2} \) |
| 89 | \( 1 - 2iT - 89T^{2} \) |
| 97 | \( 1 + 11T + 97T^{2} \) |
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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
−11.53404899071342143673972193440, −10.66518176202523632624171352051, −9.476660412762067745707152239415, −8.696753834975057074824724346127, −6.95425659078922914311067993793, −6.10830894406878056202091453064, −5.54892767092728099639388658978, −4.74896060628684584323173366671, −3.18473962013872326817762539604, −2.22851511724058885799394735378,
1.36945748312626220732479902083, 3.33671717360574232261915277963, 4.27497789850775790366433240614, 5.34026607346045504613598208763, 5.87383952946041732115373603645, 7.00172992126185058755452452244, 7.969692709288310492536008623657, 9.777626087947438429350619065172, 10.07614116417574186181496133968, 11.31964077809128755766758793830
