L(s) = 1 | + (0.587 − 0.809i)5-s + (−0.951 − 0.309i)9-s + (1.76 − 0.896i)13-s + (−0.896 + 0.142i)17-s + (−0.309 − 0.951i)25-s + (1.11 + 1.53i)29-s + (−0.809 − 1.58i)37-s + (−0.363 + 1.11i)41-s + (−0.809 + 0.587i)45-s + i·49-s + (0.309 + 0.0489i)53-s + (0.363 + 1.11i)61-s + (0.309 − 1.95i)65-s + (0.142 − 0.278i)73-s + (0.809 + 0.587i)81-s + ⋯ |
L(s) = 1 | + (0.587 − 0.809i)5-s + (−0.951 − 0.309i)9-s + (1.76 − 0.896i)13-s + (−0.896 + 0.142i)17-s + (−0.309 − 0.951i)25-s + (1.11 + 1.53i)29-s + (−0.809 − 1.58i)37-s + (−0.363 + 1.11i)41-s + (−0.809 + 0.587i)45-s + i·49-s + (0.309 + 0.0489i)53-s + (0.363 + 1.11i)61-s + (0.309 − 1.95i)65-s + (0.142 − 0.278i)73-s + (0.809 + 0.587i)81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.750 + 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.750 + 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.038146927\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.038146927\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.587 + 0.809i)T \) |
good | 3 | \( 1 + (0.951 + 0.309i)T^{2} \) |
| 7 | \( 1 - iT^{2} \) |
| 11 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 13 | \( 1 + (-1.76 + 0.896i)T + (0.587 - 0.809i)T^{2} \) |
| 17 | \( 1 + (0.896 - 0.142i)T + (0.951 - 0.309i)T^{2} \) |
| 19 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 23 | \( 1 + (-0.587 - 0.809i)T^{2} \) |
| 29 | \( 1 + (-1.11 - 1.53i)T + (-0.309 + 0.951i)T^{2} \) |
| 31 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 37 | \( 1 + (0.809 + 1.58i)T + (-0.587 + 0.809i)T^{2} \) |
| 41 | \( 1 + (0.363 - 1.11i)T + (-0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + (-0.951 - 0.309i)T^{2} \) |
| 53 | \( 1 + (-0.309 - 0.0489i)T + (0.951 + 0.309i)T^{2} \) |
| 59 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 61 | \( 1 + (-0.363 - 1.11i)T + (-0.809 + 0.587i)T^{2} \) |
| 67 | \( 1 + (0.951 - 0.309i)T^{2} \) |
| 71 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 73 | \( 1 + (-0.142 + 0.278i)T + (-0.587 - 0.809i)T^{2} \) |
| 79 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 83 | \( 1 + (-0.951 + 0.309i)T^{2} \) |
| 89 | \( 1 + (1.80 - 0.587i)T + (0.809 - 0.587i)T^{2} \) |
| 97 | \( 1 + (0.278 - 1.76i)T + (-0.951 - 0.309i)T^{2} \) |
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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
−10.54641437560242038325958480526, −9.291000955799459734747056853104, −8.674928797498916546881865912139, −8.192198531674797190113666619432, −6.68904461783371233233676843717, −5.90173597852741842246532603760, −5.20756422265877943509939286550, −3.95181827442789150578269292054, −2.80848756553467762907928258250, −1.25083029184555561704519537629,
1.89918481493679623190188232161, 3.00046108852257782490590578472, 4.10907192098185428545839539245, 5.42658034022304874998842089651, 6.34689387102231706495389691178, 6.80551206437546244944942253702, 8.251579583777890979605814305287, 8.760563975492427681426651636507, 9.788045261971032128113496986233, 10.66707562535122969719448446436