| L(s) = 1 | − 9·5-s + 5·7-s + 9·9-s − 3·11-s + 15·17-s + 19·19-s + 30·23-s + 56·25-s − 45·35-s + 85·43-s − 81·45-s − 75·47-s − 24·49-s + 27·55-s + 103·61-s + 45·63-s − 25·73-s − 15·77-s + 81·81-s − 90·83-s − 135·85-s − 171·95-s − 27·99-s − 102·101-s − 270·115-s + 75·119-s + ⋯ |
| L(s) = 1 | − 9/5·5-s + 5/7·7-s + 9-s − 0.272·11-s + 0.882·17-s + 19-s + 1.30·23-s + 2.23·25-s − 9/7·35-s + 1.97·43-s − 9/5·45-s − 1.59·47-s − 0.489·49-s + 0.490·55-s + 1.68·61-s + 5/7·63-s − 0.342·73-s − 0.194·77-s + 81-s − 1.08·83-s − 1.58·85-s − 9/5·95-s − 0.272·99-s − 1.00·101-s − 2.34·115-s + 0.630·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.389974955\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.389974955\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - p T \) |
| good | 3 | \( ( 1 - p T )( 1 + p T ) \) |
| 5 | \( 1 + 9 T + p^{2} T^{2} \) |
| 7 | \( 1 - 5 T + p^{2} T^{2} \) |
| 11 | \( 1 + 3 T + p^{2} T^{2} \) |
| 13 | \( ( 1 - p T )( 1 + p T ) \) |
| 17 | \( 1 - 15 T + p^{2} T^{2} \) |
| 23 | \( 1 - 30 T + p^{2} T^{2} \) |
| 29 | \( ( 1 - p T )( 1 + p T ) \) |
| 31 | \( ( 1 - p T )( 1 + p T ) \) |
| 37 | \( ( 1 - p T )( 1 + p T ) \) |
| 41 | \( ( 1 - p T )( 1 + p T ) \) |
| 43 | \( 1 - 85 T + p^{2} T^{2} \) |
| 47 | \( 1 + 75 T + p^{2} T^{2} \) |
| 53 | \( ( 1 - p T )( 1 + p T ) \) |
| 59 | \( ( 1 - p T )( 1 + p T ) \) |
| 61 | \( 1 - 103 T + p^{2} T^{2} \) |
| 67 | \( ( 1 - p T )( 1 + p T ) \) |
| 71 | \( ( 1 - p T )( 1 + p T ) \) |
| 73 | \( 1 + 25 T + p^{2} T^{2} \) |
| 79 | \( ( 1 - p T )( 1 + p T ) \) |
| 83 | \( 1 + 90 T + p^{2} T^{2} \) |
| 89 | \( ( 1 - p T )( 1 + p T ) \) |
| 97 | \( ( 1 - p T )( 1 + p T ) \) |
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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
−11.48526045695636845778650095183, −10.83087060790676491133142454969, −9.639925823378311079887751502701, −8.375162380748712626193591188662, −7.62815251341701531313603067890, −7.04265038386125161934451149253, −5.18654572095421912759444336950, −4.28626715473775202685433252417, −3.21949152348265669238888556293, −1.02681904521644233987753033451,
1.02681904521644233987753033451, 3.21949152348265669238888556293, 4.28626715473775202685433252417, 5.18654572095421912759444336950, 7.04265038386125161934451149253, 7.62815251341701531313603067890, 8.375162380748712626193591188662, 9.639925823378311079887751502701, 10.83087060790676491133142454969, 11.48526045695636845778650095183
