L(s) = 1 | + 5-s − 3·9-s − 4·11-s + 13-s − 6·17-s + 4·19-s + 25-s + 2·29-s + 4·31-s + 6·37-s − 6·41-s + 8·43-s − 3·45-s − 7·49-s − 2·53-s − 4·55-s + 4·59-s + 10·61-s + 65-s + 12·67-s + 4·71-s + 14·73-s + 16·79-s + 9·81-s + 12·83-s − 6·85-s + 2·89-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 9-s − 1.20·11-s + 0.277·13-s − 1.45·17-s + 0.917·19-s + 1/5·25-s + 0.371·29-s + 0.718·31-s + 0.986·37-s − 0.937·41-s + 1.21·43-s − 0.447·45-s − 49-s − 0.274·53-s − 0.539·55-s + 0.520·59-s + 1.28·61-s + 0.124·65-s + 1.46·67-s + 0.474·71-s + 1.63·73-s + 1.80·79-s + 81-s + 1.31·83-s − 0.650·85-s + 0.211·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.542685480\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.542685480\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.247815093414833943107949509171, −7.954711555661233634422659870700, −6.78881571136328219986241213377, −6.25974382144118762826388934913, −5.32912341813816679059037516143, −4.92360825859423388498808027678, −3.74834223215852470571716432157, −2.73941883514567183553958241844, −2.21993404288611753519207574379, −0.67883845902347946272676188660,
0.67883845902347946272676188660, 2.21993404288611753519207574379, 2.73941883514567183553958241844, 3.74834223215852470571716432157, 4.92360825859423388498808027678, 5.32912341813816679059037516143, 6.25974382144118762826388934913, 6.78881571136328219986241213377, 7.954711555661233634422659870700, 8.247815093414833943107949509171