L(s) = 1 | + 5-s + 11-s + 6·13-s − 3·17-s + 3·19-s − 23-s + 25-s + 7·29-s + 4·31-s − 6·37-s − 4·41-s + 9·43-s − 3·53-s + 55-s − 5·59-s − 5·61-s + 6·65-s + 2·67-s − 4·71-s + 16·73-s − 2·79-s + 3·83-s − 3·85-s − 9·89-s + 3·95-s + 7·97-s + 101-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 0.301·11-s + 1.66·13-s − 0.727·17-s + 0.688·19-s − 0.208·23-s + 1/5·25-s + 1.29·29-s + 0.718·31-s − 0.986·37-s − 0.624·41-s + 1.37·43-s − 0.412·53-s + 0.134·55-s − 0.650·59-s − 0.640·61-s + 0.744·65-s + 0.244·67-s − 0.474·71-s + 1.87·73-s − 0.225·79-s + 0.329·83-s − 0.325·85-s − 0.953·89-s + 0.307·95-s + 0.710·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.895896651\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.895896651\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−12.48410146658104, −12.02210902614880, −11.50440448028561, −11.07062976077268, −10.71761490223622, −10.21209192615991, −9.803232417286879, −9.185585280110481, −8.855372192789121, −8.448498799008846, −8.001852937632687, −7.412703141435613, −6.789496563271444, −6.424974881248962, −6.071282575424060, −5.573614111809187, −4.926628407548467, −4.522611229107841, −3.897028309141494, −3.433591219603546, −2.903353585217638, −2.295104261409240, −1.607820485836579, −1.156811422376324, −0.5407867080018531,
0.5407867080018531, 1.156811422376324, 1.607820485836579, 2.295104261409240, 2.903353585217638, 3.433591219603546, 3.897028309141494, 4.522611229107841, 4.926628407548467, 5.573614111809187, 6.071282575424060, 6.424974881248962, 6.789496563271444, 7.412703141435613, 8.001852937632687, 8.448498799008846, 8.855372192789121, 9.185585280110481, 9.803232417286879, 10.21209192615991, 10.71761490223622, 11.07062976077268, 11.50440448028561, 12.02210902614880, 12.48410146658104