| L(s) = 1 | + 6.56e3·3-s + 4.74e6·7-s + 4.30e7·9-s − 3.49e8·13-s + 3.28e10·19-s + 3.11e10·21-s + 1.52e11·25-s + 2.82e11·27-s + 1.56e12·31-s − 6.77e12·37-s − 2.29e12·39-s − 1.11e13·43-s − 1.07e13·49-s + 2.15e14·57-s + 1.84e14·61-s + 2.04e14·63-s − 7.82e14·67-s − 1.35e15·73-s + 1.00e15·75-s − 2.67e15·79-s + 1.85e15·81-s − 1.65e15·91-s + 1.02e16·93-s + 1.56e16·97-s − 2.33e16·103-s + 1.43e15·109-s − 4.44e16·111-s + ⋯ |
| L(s) = 1 | + 3-s + 0.822·7-s + 9-s − 0.428·13-s + 1.93·19-s + 0.822·21-s + 25-s + 27-s + 1.83·31-s − 1.92·37-s − 0.428·39-s − 0.956·43-s − 0.322·49-s + 1.93·57-s + 0.961·61-s + 0.822·63-s − 1.92·67-s − 1.67·73-s + 75-s − 1.76·79-s + 81-s − 0.352·91-s + 1.83·93-s + 1.99·97-s − 1.84·103-s + 0.0718·109-s − 1.92·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(17-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{17}{2})\) |
\(\approx\) |
\(3.057448288\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.057448288\) |
| \(L(9)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p^{8} T \) |
| good | 5 | \( ( 1 - p^{8} T )( 1 + p^{8} T ) \) |
| 7 | \( 1 - 4743554 T + p^{16} T^{2} \) |
| 11 | \( ( 1 - p^{8} T )( 1 + p^{8} T ) \) |
| 13 | \( 1 + 349391806 T + p^{16} T^{2} \) |
| 17 | \( ( 1 - p^{8} T )( 1 + p^{8} T ) \) |
| 19 | \( 1 - 32868566594 T + p^{16} T^{2} \) |
| 23 | \( ( 1 - p^{8} T )( 1 + p^{8} T ) \) |
| 29 | \( ( 1 - p^{8} T )( 1 + p^{8} T ) \) |
| 31 | \( 1 - 1568167997954 T + p^{16} T^{2} \) |
| 37 | \( 1 + 6771424503358 T + p^{16} T^{2} \) |
| 41 | \( ( 1 - p^{8} T )( 1 + p^{8} T ) \) |
| 43 | \( 1 + 11180981621566 T + p^{16} T^{2} \) |
| 47 | \( ( 1 - p^{8} T )( 1 + p^{8} T ) \) |
| 53 | \( ( 1 - p^{8} T )( 1 + p^{8} T ) \) |
| 59 | \( ( 1 - p^{8} T )( 1 + p^{8} T ) \) |
| 61 | \( 1 - 184288715234114 T + p^{16} T^{2} \) |
| 67 | \( 1 + 782743712096446 T + p^{16} T^{2} \) |
| 71 | \( ( 1 - p^{8} T )( 1 + p^{8} T ) \) |
| 73 | \( 1 + 1351474503392638 T + p^{16} T^{2} \) |
| 79 | \( 1 + 2677497415399678 T + p^{16} T^{2} \) |
| 83 | \( ( 1 - p^{8} T )( 1 + p^{8} T ) \) |
| 89 | \( ( 1 - p^{8} T )( 1 + p^{8} T ) \) |
| 97 | \( 1 - 15621585304991234 T + p^{16} T^{2} \) |
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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
−15.90348298685628971915936426778, −14.62232747033214551424204179423, −13.62079223435960047820523221436, −11.91411715323108263181373960703, −10.06145279156419602856755945959, −8.567302242039376965212019035700, −7.27068356794119859875953047282, −4.85479844185643493696215003270, −3.03540717343547615893953684330, −1.36640568161272534854017378942,
1.36640568161272534854017378942, 3.03540717343547615893953684330, 4.85479844185643493696215003270, 7.27068356794119859875953047282, 8.567302242039376965212019035700, 10.06145279156419602856755945959, 11.91411715323108263181373960703, 13.62079223435960047820523221436, 14.62232747033214551424204179423, 15.90348298685628971915936426778
