| L(s) = 1 | + 8.19e11·3-s + 1.12e15·4-s − 3.54e17·5-s − 4.58e22·9-s − 1.08e26·11-s + 9.23e26·12-s − 2.90e29·15-s + 1.26e30·16-s − 3.99e32·20-s + 8.08e33·23-s + 3.70e34·25-s − 6.26e35·27-s + 1.05e37·31-s − 8.88e37·33-s − 5.15e37·36-s − 2.32e39·37-s − 1.21e41·44-s + 1.62e40·45-s + 1.26e42·47-s + 1.03e42·48-s + 1.79e42·49-s + 1.86e43·53-s + 3.84e43·55-s − 4.29e43·59-s − 3.27e44·60-s + 1.42e45·64-s + 2.85e45·67-s + ⋯ |
| L(s) = 1 | + 0.967·3-s + 4-s − 1.19·5-s − 0.0638·9-s − 11-s + 0.967·12-s − 1.15·15-s + 16-s − 1.19·20-s + 0.732·23-s + 0.416·25-s − 1.02·27-s + 0.550·31-s − 0.967·33-s − 0.0638·36-s − 1.44·37-s − 44-s + 0.0759·45-s + 1.98·47-s + 0.967·48-s + 49-s + 1.45·53-s + 1.19·55-s − 0.229·59-s − 1.15·60-s + 64-s + 0.635·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(51-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+25) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{51}{2})\) |
\(\approx\) |
\(2.821407323\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.821407323\) |
| \(L(26)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + p^{25} T \) |
| good | 2 | \( ( 1 - p^{25} T )( 1 + p^{25} T ) \) |
| 3 | \( 1 - 819807014875 T + p^{50} T^{2} \) |
| 5 | \( 1 + 354715407980640001 T + p^{50} T^{2} \) |
| 7 | \( ( 1 - p^{25} T )( 1 + p^{25} T ) \) |
| 13 | \( ( 1 - p^{25} T )( 1 + p^{25} T ) \) |
| 17 | \( ( 1 - p^{25} T )( 1 + p^{25} T ) \) |
| 19 | \( ( 1 - p^{25} T )( 1 + p^{25} T ) \) |
| 23 | \( 1 - \)\(80\!\cdots\!75\)\( T + p^{50} T^{2} \) |
| 29 | \( ( 1 - p^{25} T )( 1 + p^{25} T ) \) |
| 31 | \( 1 - \)\(10\!\cdots\!23\)\( T + p^{50} T^{2} \) |
| 37 | \( 1 + \)\(23\!\cdots\!25\)\( T + p^{50} T^{2} \) |
| 41 | \( ( 1 - p^{25} T )( 1 + p^{25} T ) \) |
| 43 | \( ( 1 - p^{25} T )( 1 + p^{25} T ) \) |
| 47 | \( 1 - \)\(12\!\cdots\!50\)\( T + p^{50} T^{2} \) |
| 53 | \( 1 - \)\(18\!\cdots\!50\)\( T + p^{50} T^{2} \) |
| 59 | \( 1 + \)\(42\!\cdots\!73\)\( T + p^{50} T^{2} \) |
| 61 | \( ( 1 - p^{25} T )( 1 + p^{25} T ) \) |
| 67 | \( 1 - \)\(28\!\cdots\!75\)\( T + p^{50} T^{2} \) |
| 71 | \( 1 - \)\(33\!\cdots\!27\)\( T + p^{50} T^{2} \) |
| 73 | \( ( 1 - p^{25} T )( 1 + p^{25} T ) \) |
| 79 | \( ( 1 - p^{25} T )( 1 + p^{25} T ) \) |
| 83 | \( ( 1 - p^{25} T )( 1 + p^{25} T ) \) |
| 89 | \( 1 + \)\(10\!\cdots\!77\)\( T + p^{50} T^{2} \) |
| 97 | \( 1 - \)\(21\!\cdots\!75\)\( T + p^{50} T^{2} \) |
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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
−11.43207749957157048045265067240, −10.39864029642096940909756092740, −8.716212415679973643718100763214, −7.83176502375732848131255423977, −7.07839716944760888057893831467, −5.46910012746033023058626663936, −3.87559497853063085437414729518, −2.96956385672549890627367308004, −2.18475466455089201540143162076, −0.65790093207378628565494031657,
0.65790093207378628565494031657, 2.18475466455089201540143162076, 2.96956385672549890627367308004, 3.87559497853063085437414729518, 5.46910012746033023058626663936, 7.07839716944760888057893831467, 7.83176502375732848131255423977, 8.716212415679973643718100763214, 10.39864029642096940909756092740, 11.43207749957157048045265067240
