| L(s) = 1 | + 1.22e4·3-s + 1.04e6·4-s + 9.76e6·5-s + 2.82e8·7-s − 3.33e9·9-s − 5.18e10·11-s + 1.28e10·12-s + 9.20e10·13-s + 1.19e11·15-s + 1.09e12·16-s + 2.83e12·17-s + 1.02e13·20-s + 3.45e12·21-s + 9.53e13·25-s − 8.34e13·27-s + 2.96e14·28-s + 4.98e14·29-s − 6.33e14·33-s + 2.75e15·35-s − 3.49e15·36-s + 1.12e15·39-s − 5.43e16·44-s − 3.25e16·45-s + 1.02e17·47-s + 1.34e16·48-s + 7.97e16·49-s + 3.46e16·51-s + ⋯ |
| L(s) = 1 | + 0.206·3-s + 4-s + 5-s + 7-s − 0.957·9-s − 1.99·11-s + 0.206·12-s + 0.667·13-s + 0.206·15-s + 16-s + 1.40·17-s + 20-s + 0.206·21-s + 25-s − 0.405·27-s + 28-s + 1.18·29-s − 0.413·33-s + 35-s − 0.957·36-s + 0.138·39-s − 1.99·44-s − 0.957·45-s + 1.95·47-s + 0.206·48-s + 49-s + 0.291·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(21-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+10) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{21}{2})\) |
\(\approx\) |
\(4.089945770\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.089945770\) |
| \(L(11)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - p^{10} T \) |
| 7 | \( 1 - p^{10} T \) |
| good | 2 | \( ( 1 - p^{10} T )( 1 + p^{10} T ) \) |
| 3 | \( 1 - 12223 T + p^{20} T^{2} \) |
| 11 | \( 1 + 51836073673 T + p^{20} T^{2} \) |
| 13 | \( 1 - 92077960823 T + p^{20} T^{2} \) |
| 17 | \( 1 - 2834566008023 T + p^{20} T^{2} \) |
| 19 | \( ( 1 - p^{10} T )( 1 + p^{10} T ) \) |
| 23 | \( ( 1 - p^{10} T )( 1 + p^{10} T ) \) |
| 29 | \( 1 - 498981827484527 T + p^{20} T^{2} \) |
| 31 | \( ( 1 - p^{10} T )( 1 + p^{10} T ) \) |
| 37 | \( ( 1 - p^{10} T )( 1 + p^{10} T ) \) |
| 41 | \( ( 1 - p^{10} T )( 1 + p^{10} T ) \) |
| 43 | \( ( 1 - p^{10} T )( 1 + p^{10} T ) \) |
| 47 | \( 1 - 102681296590588223 T + p^{20} T^{2} \) |
| 53 | \( ( 1 - p^{10} T )( 1 + p^{10} T ) \) |
| 59 | \( ( 1 - p^{10} T )( 1 + p^{10} T ) \) |
| 61 | \( ( 1 - p^{10} T )( 1 + p^{10} T ) \) |
| 67 | \( ( 1 - p^{10} T )( 1 + p^{10} T ) \) |
| 71 | \( 1 + 6446014172213061598 T + p^{20} T^{2} \) |
| 73 | \( 1 - 6042116169407251298 T + p^{20} T^{2} \) |
| 79 | \( 1 - 8153339628309741527 T + p^{20} T^{2} \) |
| 83 | \( 1 + 20776721187438397102 T + p^{20} T^{2} \) |
| 89 | \( ( 1 - p^{10} T )( 1 + p^{10} T ) \) |
| 97 | \( 1 - \)\(11\!\cdots\!23\)\( T + p^{20} T^{2} \) |
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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
−12.22864855121993251016889046121, −10.92725093422851985775331831977, −10.25065298819424541248500033511, −8.458975729454926479197897129068, −7.58459731918162193963430213394, −5.90217313703160996565606459257, −5.25174346514711508375942454328, −2.98086009873935343015630722746, −2.25677947589895626979057469968, −1.02954942553762909601684236388,
1.02954942553762909601684236388, 2.25677947589895626979057469968, 2.98086009873935343015630722746, 5.25174346514711508375942454328, 5.90217313703160996565606459257, 7.58459731918162193963430213394, 8.458975729454926479197897129068, 10.25065298819424541248500033511, 10.92725093422851985775331831977, 12.22864855121993251016889046121
