L(s) = 1 | + (1 − 1.41i)3-s + (−2.23 + 1.41i)7-s + (−1.00 − 2.82i)9-s + 2.82i·11-s + 2.82i·13-s + 4·17-s − 6.32i·19-s + (−0.236 + 4.57i)21-s − 6.32i·23-s + (−5.00 − 1.41i)27-s − 5.65i·29-s − 6.32i·31-s + (4.00 + 2.82i)33-s + 8.94·37-s + (4.00 + 2.82i)39-s + ⋯ |
L(s) = 1 | + (0.577 − 0.816i)3-s + (−0.845 + 0.534i)7-s + (−0.333 − 0.942i)9-s + 0.852i·11-s + 0.784i·13-s + 0.970·17-s − 1.45i·19-s + (−0.0515 + 0.998i)21-s − 1.31i·23-s + (−0.962 − 0.272i)27-s − 1.05i·29-s − 1.13i·31-s + (0.696 + 0.492i)33-s + 1.47·37-s + (0.640 + 0.452i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0515 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0515 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.704201034\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.704201034\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1 + 1.41i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2.23 - 1.41i)T \) |
good | 11 | \( 1 - 2.82iT - 11T^{2} \) |
| 13 | \( 1 - 2.82iT - 13T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 + 6.32iT - 19T^{2} \) |
| 23 | \( 1 + 6.32iT - 23T^{2} \) |
| 29 | \( 1 + 5.65iT - 29T^{2} \) |
| 31 | \( 1 + 6.32iT - 31T^{2} \) |
| 37 | \( 1 - 8.94T + 37T^{2} \) |
| 41 | \( 1 + 4.47T + 41T^{2} \) |
| 43 | \( 1 - 4.47T + 43T^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 + 6.32iT - 53T^{2} \) |
| 59 | \( 1 + 8.94T + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 4.47T + 67T^{2} \) |
| 71 | \( 1 + 14.1iT - 71T^{2} \) |
| 73 | \( 1 + 8.48iT - 73T^{2} \) |
| 79 | \( 1 - 12T + 79T^{2} \) |
| 83 | \( 1 + 10T + 83T^{2} \) |
| 89 | \( 1 + 4.47T + 89T^{2} \) |
| 97 | \( 1 - 8.48iT - 97T^{2} \) |
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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
−9.093726516102119278882378030508, −8.023556842419293705497796831902, −7.38602873218251697817370182729, −6.52010210179974891981294856239, −6.10761152260331602636290958478, −4.81588145847436760520036011753, −3.85421372371903882723711967465, −2.69728818112346763175590312435, −2.17486628800189304740489406178, −0.59906208980625665939897346550,
1.25301348025842130937741264451, 2.98750901658983860713578081899, 3.38632192914247224779042283096, 4.18664215981407121322338641075, 5.50160908928694922955219877676, 5.83505520196398056942323751083, 7.16260864929256264513932708125, 7.87747048949143080415613415559, 8.532258564263858957899227098985, 9.452834439660390204373402312738