| L(s) = 1 | + i·3-s − i·5-s − 3.79·7-s − 9-s − 3.69·11-s + 2.60·13-s + 15-s + 1.60i·17-s + 8.20·19-s − 3.79i·21-s + (−1.19 + 4.64i)23-s − 25-s − i·27-s + 0.706·29-s + 6.46i·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 0.447i·5-s − 1.43·7-s − 0.333·9-s − 1.11·11-s + 0.723·13-s + 0.258·15-s + 0.388i·17-s + 1.88·19-s − 0.829i·21-s + (−0.249 + 0.968i)23-s − 0.200·25-s − 0.192i·27-s + 0.131·29-s + 1.16i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.249 + 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.249 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4788613000\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4788613000\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + iT \) |
| 23 | \( 1 + (1.19 - 4.64i)T \) |
| good | 7 | \( 1 + 3.79T + 7T^{2} \) |
| 11 | \( 1 + 3.69T + 11T^{2} \) |
| 13 | \( 1 - 2.60T + 13T^{2} \) |
| 17 | \( 1 - 1.60iT - 17T^{2} \) |
| 19 | \( 1 - 8.20T + 19T^{2} \) |
| 29 | \( 1 - 0.706T + 29T^{2} \) |
| 31 | \( 1 - 6.46iT - 31T^{2} \) |
| 37 | \( 1 + 5.01iT - 37T^{2} \) |
| 41 | \( 1 + 10.6T + 41T^{2} \) |
| 43 | \( 1 - 5.94T + 43T^{2} \) |
| 47 | \( 1 - 4.46iT - 47T^{2} \) |
| 53 | \( 1 + 7.25iT - 53T^{2} \) |
| 59 | \( 1 + 6.04iT - 59T^{2} \) |
| 61 | \( 1 - 3.36iT - 61T^{2} \) |
| 67 | \( 1 + 13.5T + 67T^{2} \) |
| 71 | \( 1 - 8.69iT - 71T^{2} \) |
| 73 | \( 1 + 0.516T + 73T^{2} \) |
| 79 | \( 1 - 3.23T + 79T^{2} \) |
| 83 | \( 1 + 4.10T + 83T^{2} \) |
| 89 | \( 1 + 5.11iT - 89T^{2} \) |
| 97 | \( 1 + 19.4iT - 97T^{2} \) |
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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
−7.969790721773456126662463668391, −7.27868564845527680588308992231, −6.44608167228426092259724149654, −5.53694808166608285243609720554, −5.30872201355496426332869699412, −4.14890672558419332914141054309, −3.31616117623929404741007636883, −2.95293136453329987459647990211, −1.48245417556115052229603136653, −0.14845652325063413522278762508,
0.952913631556006715234255705619, 2.38453283549730516525822105149, 3.02780637570519410685001794559, 3.60180618422998458514311299910, 4.82957100926029090512063908076, 5.73555442638380183259520149342, 6.22596128653198595549174242151, 6.96680616671236578118457350470, 7.54540279294268379154569767820, 8.204494989840319960275288105005
