| L(s) = 1 | + 2.24·3-s + 0.246·5-s + 2.35·7-s + 2.04·9-s + 4.24·11-s + 0.554·15-s + 2.15·17-s + 0.0881·19-s + 5.29·21-s − 1.49·23-s − 4.93·25-s − 2.13·27-s + 4.63·29-s + 6.63·31-s + 9.54·33-s + 0.582·35-s + 5.69·37-s − 11.5·41-s + 0.295·43-s + 0.506·45-s + 7.35·47-s − 1.44·49-s + 4.85·51-s − 10.3·53-s + 1.04·55-s + 0.198·57-s + 6.78·59-s + ⋯ |
| L(s) = 1 | + 1.29·3-s + 0.110·5-s + 0.890·7-s + 0.682·9-s + 1.28·11-s + 0.143·15-s + 0.523·17-s + 0.0202·19-s + 1.15·21-s − 0.311·23-s − 0.987·25-s − 0.411·27-s + 0.859·29-s + 1.19·31-s + 1.66·33-s + 0.0983·35-s + 0.935·37-s − 1.81·41-s + 0.0451·43-s + 0.0754·45-s + 1.07·47-s − 0.206·49-s + 0.679·51-s − 1.42·53-s + 0.141·55-s + 0.0262·57-s + 0.882·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.599450317\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.599450317\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 2.24T + 3T^{2} \) |
| 5 | \( 1 - 0.246T + 5T^{2} \) |
| 7 | \( 1 - 2.35T + 7T^{2} \) |
| 11 | \( 1 - 4.24T + 11T^{2} \) |
| 17 | \( 1 - 2.15T + 17T^{2} \) |
| 19 | \( 1 - 0.0881T + 19T^{2} \) |
| 23 | \( 1 + 1.49T + 23T^{2} \) |
| 29 | \( 1 - 4.63T + 29T^{2} \) |
| 31 | \( 1 - 6.63T + 31T^{2} \) |
| 37 | \( 1 - 5.69T + 37T^{2} \) |
| 41 | \( 1 + 11.5T + 41T^{2} \) |
| 43 | \( 1 - 0.295T + 43T^{2} \) |
| 47 | \( 1 - 7.35T + 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 - 6.78T + 59T^{2} \) |
| 61 | \( 1 - 3.47T + 61T^{2} \) |
| 67 | \( 1 + 7.67T + 67T^{2} \) |
| 71 | \( 1 - 8.66T + 71T^{2} \) |
| 73 | \( 1 - 6.73T + 73T^{2} \) |
| 79 | \( 1 + 9.97T + 79T^{2} \) |
| 83 | \( 1 + 1.60T + 83T^{2} \) |
| 89 | \( 1 + 2.88T + 89T^{2} \) |
| 97 | \( 1 + 8.05T + 97T^{2} \) |
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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
−8.558200668972646953981682575503, −8.316963004057204357769601651046, −7.55814399078563229109632091398, −6.66126811803872301237941351377, −5.80448262878986490786631528333, −4.68880637238891790344981114107, −3.95861882341928241832320521658, −3.12857659046338928237243874503, −2.12416935148308569156316907932, −1.26535983341148795985608467855,
1.26535983341148795985608467855, 2.12416935148308569156316907932, 3.12857659046338928237243874503, 3.95861882341928241832320521658, 4.68880637238891790344981114107, 5.80448262878986490786631528333, 6.66126811803872301237941351377, 7.55814399078563229109632091398, 8.316963004057204357769601651046, 8.558200668972646953981682575503
