L(s) = 1 | + 11-s + 2·13-s + 2·17-s + 4·19-s − 5·25-s + 6·29-s + 2·31-s − 2·37-s − 6·41-s − 6·47-s − 2·53-s + 2·59-s − 2·61-s − 4·67-s − 6·73-s + 12·79-s − 4·83-s + 8·89-s + 12·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
L(s) = 1 | + 0.301·11-s + 0.554·13-s + 0.485·17-s + 0.917·19-s − 25-s + 1.11·29-s + 0.359·31-s − 0.328·37-s − 0.937·41-s − 0.875·47-s − 0.274·53-s + 0.260·59-s − 0.256·61-s − 0.488·67-s − 0.702·73-s + 1.35·79-s − 0.439·83-s + 0.847·89-s + 1.21·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38808 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.514931468\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.514931468\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−14.84668301352871, −14.13726819759417, −13.86765379130687, −13.35843977316015, −12.74172011900801, −12.11095980017554, −11.66011004253973, −11.35754956484492, −10.43029916155578, −10.15116663375399, −9.562804115156188, −8.940475780228495, −8.425880381143168, −7.809765190478768, −7.369586042285162, −6.538503380346363, −6.205014802509760, −5.465206746977730, −4.904989984492494, −4.247454177912366, −3.433971540628066, −3.127003965931572, −2.099407843399749, −1.404023873663841, −0.6121399461500174,
0.6121399461500174, 1.404023873663841, 2.099407843399749, 3.127003965931572, 3.433971540628066, 4.247454177912366, 4.904989984492494, 5.465206746977730, 6.205014802509760, 6.538503380346363, 7.369586042285162, 7.809765190478768, 8.425880381143168, 8.940475780228495, 9.562804115156188, 10.15116663375399, 10.43029916155578, 11.35754956484492, 11.66011004253973, 12.11095980017554, 12.74172011900801, 13.35843977316015, 13.86765379130687, 14.13726819759417, 14.84668301352871