| L(s) = 1 | + 6·11-s − 6·17-s + 3·19-s + 6·23-s − 5·25-s − 9·29-s − 6·31-s + 9·37-s + 3·41-s + 8·43-s + 3·47-s − 7·49-s + 6·53-s − 8·61-s + 12·67-s − 3·71-s − 6·73-s − 79-s − 6·83-s + 15·89-s + 12·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 1.80·11-s − 1.45·17-s + 0.688·19-s + 1.25·23-s − 25-s − 1.67·29-s − 1.07·31-s + 1.47·37-s + 0.468·41-s + 1.21·43-s + 0.437·47-s − 49-s + 0.824·53-s − 1.02·61-s + 1.46·67-s − 0.356·71-s − 0.702·73-s − 0.112·79-s − 0.658·83-s + 1.58·89-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 73008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.514833737\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.514833737\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 9 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 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
−14.31217322261334, −13.54319011979495, −13.00326817883393, −12.81401169297335, −11.94219860869508, −11.52362514469613, −11.17156146183344, −10.81511578470498, −9.896190317042841, −9.356834778212398, −9.061460101879865, −8.820246966909384, −7.693919090078993, −7.522032577451822, −6.805817927451687, −6.357170188421331, −5.808280458378356, −5.216776222899749, −4.399421623555761, −4.009164216327269, −3.526571249476182, −2.668966020584394, −1.966417548870183, −1.349732755482404, −0.5428120763730696,
0.5428120763730696, 1.349732755482404, 1.966417548870183, 2.668966020584394, 3.526571249476182, 4.009164216327269, 4.399421623555761, 5.216776222899749, 5.808280458378356, 6.357170188421331, 6.805817927451687, 7.522032577451822, 7.693919090078993, 8.820246966909384, 9.061460101879865, 9.356834778212398, 9.896190317042841, 10.81511578470498, 11.17156146183344, 11.52362514469613, 11.94219860869508, 12.81401169297335, 13.00326817883393, 13.54319011979495, 14.31217322261334
