| L(s) = 1 | − 3.49e5·3-s + 4.19e6·4-s + 8.71e7·5-s + 9.09e10·9-s − 2.85e11·11-s − 1.46e12·12-s − 3.04e13·15-s + 1.75e13·16-s + 3.65e14·20-s + 1.64e14·23-s + 5.20e15·25-s − 2.08e16·27-s + 3.46e16·31-s + 9.98e16·33-s + 3.81e17·36-s − 2.15e17·37-s − 1.19e18·44-s + 7.92e18·45-s + 5.61e17·47-s − 6.15e18·48-s + 3.90e18·49-s + 1.84e19·53-s − 2.48e19·55-s + 4.51e18·59-s − 1.27e20·60-s + 7.37e19·64-s − 5.68e19·67-s + ⋯ |
| L(s) = 1 | − 1.97·3-s + 4-s + 1.78·5-s + 2.89·9-s − 11-s − 1.97·12-s − 3.52·15-s + 16-s + 1.78·20-s + 0.172·23-s + 2.18·25-s − 3.75·27-s + 1.36·31-s + 1.97·33-s + 2.89·36-s − 1.20·37-s − 44-s + 5.17·45-s + 0.226·47-s − 1.97·48-s + 49-s + 1.99·53-s − 1.78·55-s + 0.149·59-s − 3.52·60-s + 64-s − 0.465·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(23-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+11) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{23}{2})\) |
\(\approx\) |
\(1.893569989\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.893569989\) |
| \(L(12)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + p^{11} T \) |
| good | 2 | \( ( 1 - p^{11} T )( 1 + p^{11} T ) \) |
| 3 | \( 1 + 349805 T + p^{22} T^{2} \) |
| 5 | \( 1 - 87113399 T + p^{22} T^{2} \) |
| 7 | \( ( 1 - p^{11} T )( 1 + p^{11} T ) \) |
| 13 | \( ( 1 - p^{11} T )( 1 + p^{11} T ) \) |
| 17 | \( ( 1 - p^{11} T )( 1 + p^{11} T ) \) |
| 19 | \( ( 1 - p^{11} T )( 1 + p^{11} T ) \) |
| 23 | \( 1 - 164431835937635 T + p^{22} T^{2} \) |
| 29 | \( ( 1 - p^{11} T )( 1 + p^{11} T ) \) |
| 31 | \( 1 - 34677411476888363 T + p^{22} T^{2} \) |
| 37 | \( 1 + 215123718503529025 T + p^{22} T^{2} \) |
| 41 | \( ( 1 - p^{11} T )( 1 + p^{11} T ) \) |
| 43 | \( ( 1 - p^{11} T )( 1 + p^{11} T ) \) |
| 47 | \( 1 - 561066715837848050 T + p^{22} T^{2} \) |
| 53 | \( 1 - 18477962235892882730 T + p^{22} T^{2} \) |
| 59 | \( 1 - 4517320382583181907 T + p^{22} T^{2} \) |
| 61 | \( ( 1 - p^{11} T )( 1 + p^{11} T ) \) |
| 67 | \( 1 + 56813482553573915365 T + p^{22} T^{2} \) |
| 71 | \( 1 - \)\(32\!\cdots\!67\)\( T + p^{22} T^{2} \) |
| 73 | \( ( 1 - p^{11} T )( 1 + p^{11} T ) \) |
| 79 | \( ( 1 - p^{11} T )( 1 + p^{11} T ) \) |
| 83 | \( ( 1 - p^{11} T )( 1 + p^{11} T ) \) |
| 89 | \( 1 - \)\(31\!\cdots\!03\)\( T + p^{22} T^{2} \) |
| 97 | \( 1 - \)\(87\!\cdots\!95\)\( T + p^{22} T^{2} \) |
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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
−15.59425271013570022862496981813, −13.31797464395066843890859709387, −12.11398813851925343457675728366, −10.68333685719000497142004214235, −10.08675215252931750681483524401, −6.95837770434728255522504554909, −5.96309498378690411368853287393, −5.18663428142283367028067241389, −2.18808149600072433564311818092, −0.967055379028228794397480231861,
0.967055379028228794397480231861, 2.18808149600072433564311818092, 5.18663428142283367028067241389, 5.96309498378690411368853287393, 6.95837770434728255522504554909, 10.08675215252931750681483524401, 10.68333685719000497142004214235, 12.11398813851925343457675728366, 13.31797464395066843890859709387, 15.59425271013570022862496981813
