| L(s) = 1 | + 9·3-s + 16·4-s + 23·7-s + 81·9-s + 144·12-s − 337·13-s + 256·16-s + 647·19-s + 207·21-s + 729·27-s + 368·28-s − 1.75e3·31-s + 1.29e3·36-s − 2.06e3·37-s − 3.03e3·39-s + 23·43-s + 2.30e3·48-s − 1.87e3·49-s − 5.39e3·52-s + 5.82e3·57-s − 5.23e3·61-s + 1.86e3·63-s + 4.09e3·64-s + 2.90e3·67-s − 8.54e3·73-s + 1.03e4·76-s + 7.68e3·79-s + ⋯ |
| L(s) = 1 | + 3-s + 4-s + 0.469·7-s + 9-s + 12-s − 1.99·13-s + 16-s + 1.79·19-s + 0.469·21-s + 27-s + 0.469·28-s − 1.82·31-s + 36-s − 1.50·37-s − 1.99·39-s + 0.0124·43-s + 48-s − 0.779·49-s − 1.99·52-s + 1.79·57-s − 1.40·61-s + 0.469·63-s + 64-s + 0.646·67-s − 1.60·73-s + 1.79·76-s + 1.23·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.780628261\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.780628261\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - p^{2} T \) |
| 5 | \( 1 \) |
| good | 2 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 7 | \( 1 - 23 T + p^{4} T^{2} \) |
| 11 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 13 | \( 1 + 337 T + p^{4} T^{2} \) |
| 17 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 19 | \( 1 - 647 T + p^{4} T^{2} \) |
| 23 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 29 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 31 | \( 1 + 1753 T + p^{4} T^{2} \) |
| 37 | \( 1 + 2062 T + p^{4} T^{2} \) |
| 41 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 43 | \( 1 - 23 T + p^{4} T^{2} \) |
| 47 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 53 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 59 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 61 | \( 1 + 5233 T + p^{4} T^{2} \) |
| 67 | \( 1 - 2903 T + p^{4} T^{2} \) |
| 71 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 73 | \( 1 + 8542 T + p^{4} T^{2} \) |
| 79 | \( 1 - 7682 T + p^{4} T^{2} \) |
| 83 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 89 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 97 | \( 1 - 9743 T + p^{4} T^{2} \) |
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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
−14.11871909382922466647228281821, −12.61251511130612925540717230837, −11.72130387472548392917208685514, −10.32323946298088375737990587463, −9.310310356846916737252968348194, −7.69906588291310245697888529564, −7.16247687597561587119231272442, −5.14021592860931460246567086610, −3.18310784597666974750276484199, −1.86773543549791273166421485733,
1.86773543549791273166421485733, 3.18310784597666974750276484199, 5.14021592860931460246567086610, 7.16247687597561587119231272442, 7.69906588291310245697888529564, 9.310310356846916737252968348194, 10.32323946298088375737990587463, 11.72130387472548392917208685514, 12.61251511130612925540717230837, 14.11871909382922466647228281821
