L(s) = 1 | + 2.44i·3-s + (−2.22 + 0.224i)5-s + i·7-s − 2.99·9-s − 4.89·11-s + 4.44i·13-s + (−0.550 − 5.44i)15-s + 2i·17-s − 1.55·19-s − 2.44·21-s + 2.89i·23-s + (4.89 − i)25-s + 6.89·29-s − 8.89·31-s − 11.9i·33-s + ⋯ |
L(s) = 1 | + 1.41i·3-s + (−0.994 + 0.100i)5-s + 0.377i·7-s − 0.999·9-s − 1.47·11-s + 1.23i·13-s + (−0.142 − 1.40i)15-s + 0.485i·17-s − 0.355·19-s − 0.534·21-s + 0.604i·23-s + (0.979 − 0.200i)25-s + 1.28·29-s − 1.59·31-s − 2.08i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.100 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.100 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2342096602\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2342096602\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (2.22 - 0.224i)T \) |
| 7 | \( 1 - iT \) |
good | 3 | \( 1 - 2.44iT - 3T^{2} \) |
| 11 | \( 1 + 4.89T + 11T^{2} \) |
| 13 | \( 1 - 4.44iT - 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 + 1.55T + 19T^{2} \) |
| 23 | \( 1 - 2.89iT - 23T^{2} \) |
| 29 | \( 1 - 6.89T + 29T^{2} \) |
| 31 | \( 1 + 8.89T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 1.10T + 41T^{2} \) |
| 43 | \( 1 - 0.898iT - 43T^{2} \) |
| 47 | \( 1 + 8.89iT - 47T^{2} \) |
| 53 | \( 1 + 10.8iT - 53T^{2} \) |
| 59 | \( 1 - 1.55T + 59T^{2} \) |
| 61 | \( 1 + 3.55T + 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 - 1.10T + 71T^{2} \) |
| 73 | \( 1 + 2.89iT - 73T^{2} \) |
| 79 | \( 1 - 6.89T + 79T^{2} \) |
| 83 | \( 1 - 2.44iT - 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 - 15.7iT - 97T^{2} \) |
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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
−9.623715931130293094885658048634, −8.883170383829344345756038598822, −8.265493197091259019378217084591, −7.43467998397926474472214561327, −6.50804722989504676949278244825, −5.30904086575016875603804969980, −4.83168752410044702649472047185, −3.93740943058988220453295133434, −3.33740105094551033692078193691, −2.15483673423177973115574958474,
0.094002866957316846688170807967, 1.01771447463130669325798614860, 2.48961387616786359637251526014, 3.16540253905723505462947224391, 4.46304683750416693547648525364, 5.31861202553665876114168051871, 6.25154924433436304318965622659, 7.19534708049783395195133406631, 7.65126835033325308410922788456, 8.111159900199570790743225068152