L(s) = 1 | + (2 − i)5-s + i·7-s + 4i·13-s − 2i·17-s − 8·19-s + 8i·23-s + (3 − 4i)25-s + 8·29-s − 4·31-s + (1 + 2i)35-s + 8i·37-s − 12·41-s − 8i·43-s + 4i·47-s − 49-s + ⋯ |
L(s) = 1 | + (0.894 − 0.447i)5-s + 0.377i·7-s + 1.10i·13-s − 0.485i·17-s − 1.83·19-s + 1.66i·23-s + (0.600 − 0.800i)25-s + 1.48·29-s − 0.718·31-s + (0.169 + 0.338i)35-s + 1.31i·37-s − 1.87·41-s − 1.21i·43-s + 0.583i·47-s − 0.142·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.266757961\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.266757961\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-2 + i)T \) |
| 7 | \( 1 - iT \) |
good | 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 + 8T + 19T^{2} \) |
| 23 | \( 1 - 8iT - 23T^{2} \) |
| 29 | \( 1 - 8T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 8iT - 37T^{2} \) |
| 41 | \( 1 + 12T + 41T^{2} \) |
| 43 | \( 1 + 8iT - 43T^{2} \) |
| 47 | \( 1 - 4iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 4iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + 4T + 89T^{2} \) |
| 97 | \( 1 - 12iT - 97T^{2} \) |
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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
−8.687034078389888599956455145137, −7.85003515183107985585570864504, −6.74201679680217856153555134743, −6.44348308624615071114584636469, −5.52558682272690086878815941010, −4.86064859011381207697476780291, −4.14882272281692249035156862655, −3.02303042816921714036285258224, −2.06551600039649436516806368818, −1.41317814317771354390157819295,
0.30863265922753188692794668314, 1.69386519481983963139667836046, 2.52539858277502070462823824253, 3.33516454100800292085131580258, 4.35364185073511004080108465126, 5.06971579066216886746493290452, 6.03382695490787859462403850567, 6.46979954200226927275397860038, 7.11649220416291560652323164152, 8.255991205840487765525451927045