L(s) = 1 | + (1.36 − 0.351i)2-s − 3-s + (1.75 − 0.962i)4-s − 3.58i·5-s + (−1.36 + 0.351i)6-s + 4.07·7-s + (2.06 − 1.93i)8-s + 9-s + (−1.25 − 4.91i)10-s − 4.30·11-s + (−1.75 + 0.962i)12-s − 0.588i·13-s + (5.58 − 1.43i)14-s + 3.58i·15-s + (2.14 − 3.37i)16-s + 0.330·17-s + ⋯ |
L(s) = 1 | + (0.968 − 0.248i)2-s − 0.577·3-s + (0.876 − 0.481i)4-s − 1.60i·5-s + (−0.559 + 0.143i)6-s + 1.53·7-s + (0.729 − 0.683i)8-s + 0.333·9-s + (−0.398 − 1.55i)10-s − 1.29·11-s + (−0.506 + 0.277i)12-s − 0.163i·13-s + (1.49 − 0.382i)14-s + 0.926i·15-s + (0.537 − 0.843i)16-s + 0.0801·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0566 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0566 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.78078 - 1.88477i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.78078 - 1.88477i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.36 + 0.351i)T \) |
| 3 | \( 1 + T \) |
| 67 | \( 1 + (-4.33 + 6.94i)T \) |
good | 5 | \( 1 + 3.58iT - 5T^{2} \) |
| 7 | \( 1 - 4.07T + 7T^{2} \) |
| 11 | \( 1 + 4.30T + 11T^{2} \) |
| 13 | \( 1 + 0.588iT - 13T^{2} \) |
| 17 | \( 1 - 0.330T + 17T^{2} \) |
| 19 | \( 1 - 5.25iT - 19T^{2} \) |
| 23 | \( 1 + 0.501iT - 23T^{2} \) |
| 29 | \( 1 + 5.91T + 29T^{2} \) |
| 31 | \( 1 - 5.74T + 31T^{2} \) |
| 37 | \( 1 + 1.03T + 37T^{2} \) |
| 41 | \( 1 + 7.04iT - 41T^{2} \) |
| 43 | \( 1 - 6.86T + 43T^{2} \) |
| 47 | \( 1 - 10.3iT - 47T^{2} \) |
| 53 | \( 1 - 8.47iT - 53T^{2} \) |
| 59 | \( 1 + 8.93iT - 59T^{2} \) |
| 61 | \( 1 + 6.57iT - 61T^{2} \) |
| 71 | \( 1 - 0.0930iT - 71T^{2} \) |
| 73 | \( 1 + 6.32T + 73T^{2} \) |
| 79 | \( 1 + 1.43T + 79T^{2} \) |
| 83 | \( 1 - 12.5iT - 83T^{2} \) |
| 89 | \( 1 - 18.3T + 89T^{2} \) |
| 97 | \( 1 - 6.83iT - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.41853702811164058765555243098, −9.263326353092201819738853660175, −7.951859919476437450210094231641, −7.75737237673896341226768217499, −6.02173729875720581962177385034, −5.24419741673969889647327984347, −4.88329697959021047579072576663, −3.97860652759347017685081705833, −2.12984622713311344348074902988, −1.08917398874196517166722722911,
2.09452040863470249340323657428, 2.97920631599108142835721022972, 4.33578686289813008562996591568, 5.17159709547168716087451535593, 5.96198234858967713836278773235, 7.07081052231366629625195367803, 7.47415678564901506962605462717, 8.398860926836578510906270992022, 10.16713515595551284483967187681, 10.77388905142531004156884110826