L(s) = 1 | + (−1.21 − 0.723i)2-s + 3-s + (0.952 + 1.75i)4-s + 4.31·5-s + (−1.21 − 0.723i)6-s − 1.64·7-s + (0.114 − 2.82i)8-s + 9-s + (−5.23 − 3.11i)10-s − 2.66i·11-s + (0.952 + 1.75i)12-s − 4.33i·13-s + (2.00 + 1.19i)14-s + 4.31·15-s + (−2.18 + 3.35i)16-s − 0.417i·17-s + ⋯ |
L(s) = 1 | + (−0.859 − 0.511i)2-s + 0.577·3-s + (0.476 + 0.879i)4-s + 1.92·5-s + (−0.496 − 0.295i)6-s − 0.622·7-s + (0.0406 − 0.999i)8-s + 0.333·9-s + (−1.65 − 0.986i)10-s − 0.803i·11-s + (0.275 + 0.507i)12-s − 1.20i·13-s + (0.534 + 0.318i)14-s + 1.11·15-s + (−0.546 + 0.837i)16-s − 0.101i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.626 + 0.779i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.626 + 0.779i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.38737 - 0.664702i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.38737 - 0.664702i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.21 + 0.723i)T \) |
| 3 | \( 1 - T \) |
| 23 | \( 1 + (3.61 - 3.15i)T \) |
good | 5 | \( 1 - 4.31T + 5T^{2} \) |
| 7 | \( 1 + 1.64T + 7T^{2} \) |
| 11 | \( 1 + 2.66iT - 11T^{2} \) |
| 13 | \( 1 + 4.33iT - 13T^{2} \) |
| 17 | \( 1 + 0.417iT - 17T^{2} \) |
| 19 | \( 1 + 7.23iT - 19T^{2} \) |
| 29 | \( 1 - 9.62iT - 29T^{2} \) |
| 31 | \( 1 - 6.72iT - 31T^{2} \) |
| 37 | \( 1 - 5.50T + 37T^{2} \) |
| 41 | \( 1 - 1.83T + 41T^{2} \) |
| 43 | \( 1 - 1.68iT - 43T^{2} \) |
| 47 | \( 1 - 1.93iT - 47T^{2} \) |
| 53 | \( 1 + 5.44T + 53T^{2} \) |
| 59 | \( 1 - 9.10T + 59T^{2} \) |
| 61 | \( 1 - 0.402T + 61T^{2} \) |
| 67 | \( 1 + 5.42iT - 67T^{2} \) |
| 71 | \( 1 - 1.27iT - 71T^{2} \) |
| 73 | \( 1 + 8.90T + 73T^{2} \) |
| 79 | \( 1 + 11.1T + 79T^{2} \) |
| 83 | \( 1 - 11.0iT - 83T^{2} \) |
| 89 | \( 1 - 18.4iT - 89T^{2} \) |
| 97 | \( 1 + 9.76iT - 97T^{2} \) |
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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
−10.44058134852472201745628414376, −9.676824205379024479971445540409, −9.128492092067523255524782475489, −8.385626243055948697207071326532, −7.09222912812362388467695051966, −6.26963447986069693368190531651, −5.19019617983323570132432914818, −3.20473257660112865892991231006, −2.63212252386416197173083828236, −1.23308220328723258846936593035,
1.75296369283826651595076513054, 2.38312127406360719050131184517, 4.39895547456215034548374423916, 5.95975576863590773589252637851, 6.22552665844251887337296374331, 7.32494728530145721712366910064, 8.425419182187225102197787823630, 9.368270002476629393719043659923, 9.945073246177348801767427382713, 10.11986636102439067646783398036