| L(s) = 1 | − 3.41e5·3-s + 4.19e6·4-s − 4.88e7·5-s + 1.97e9·7-s + 8.51e10·9-s + 3.54e11·11-s − 1.43e12·12-s + 1.43e12·13-s + 1.66e13·15-s + 1.75e13·16-s + 6.65e13·17-s − 2.04e14·20-s − 6.74e14·21-s + 2.38e15·25-s − 1.83e16·27-s + 8.29e15·28-s + 2.37e16·29-s − 1.21e17·33-s − 9.65e16·35-s + 3.57e17·36-s − 4.88e17·39-s + 1.48e18·44-s − 4.15e18·45-s − 2.60e18·47-s − 6.00e18·48-s + 3.90e18·49-s − 2.27e19·51-s + ⋯ |
| L(s) = 1 | − 1.92·3-s + 4-s − 5-s + 7-s + 2.71·9-s + 1.24·11-s − 1.92·12-s + 0.798·13-s + 1.92·15-s + 16-s + 1.94·17-s − 20-s − 1.92·21-s + 25-s − 3.30·27-s + 28-s + 1.94·29-s − 2.39·33-s − 35-s + 2.71·36-s − 1.53·39-s + 1.24·44-s − 2.71·45-s − 1.05·47-s − 1.92·48-s + 49-s − 3.74·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(23-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+11) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{23}{2})\) |
\(\approx\) |
\(2.061648937\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.061648937\) |
| \(L(12)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + p^{11} T \) |
| 7 | \( 1 - p^{11} T \) |
| good | 2 | \( ( 1 - p^{11} T )( 1 + p^{11} T ) \) |
| 3 | \( 1 + 341351 T + p^{22} T^{2} \) |
| 11 | \( 1 - 354730232987 T + p^{22} T^{2} \) |
| 13 | \( 1 - 1431532005269 T + p^{22} T^{2} \) |
| 17 | \( 1 - 66547133948621 T + p^{22} T^{2} \) |
| 19 | \( ( 1 - p^{11} T )( 1 + p^{11} T ) \) |
| 23 | \( ( 1 - p^{11} T )( 1 + p^{11} T ) \) |
| 29 | \( 1 - 23774726872835423 T + p^{22} T^{2} \) |
| 31 | \( ( 1 - p^{11} T )( 1 + p^{11} T ) \) |
| 37 | \( ( 1 - p^{11} T )( 1 + p^{11} T ) \) |
| 41 | \( ( 1 - p^{11} T )( 1 + p^{11} T ) \) |
| 43 | \( ( 1 - p^{11} T )( 1 + p^{11} T ) \) |
| 47 | \( 1 + 2606499276897091519 T + p^{22} T^{2} \) |
| 53 | \( ( 1 - p^{11} T )( 1 + p^{11} T ) \) |
| 59 | \( ( 1 - p^{11} T )( 1 + p^{11} T ) \) |
| 61 | \( ( 1 - p^{11} T )( 1 + p^{11} T ) \) |
| 67 | \( ( 1 - p^{11} T )( 1 + p^{11} T ) \) |
| 71 | \( 1 + 71331542064478634398 T + p^{22} T^{2} \) |
| 73 | \( 1 - \)\(33\!\cdots\!34\)\( T + p^{22} T^{2} \) |
| 79 | \( 1 + \)\(48\!\cdots\!57\)\( T + p^{22} T^{2} \) |
| 83 | \( 1 - \)\(74\!\cdots\!14\)\( T + p^{22} T^{2} \) |
| 89 | \( ( 1 - p^{11} T )( 1 + p^{11} T ) \) |
| 97 | \( 1 - \)\(29\!\cdots\!01\)\( T + p^{22} T^{2} \) |
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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
−11.85095090606809206981459199253, −11.17952798472451000381609481305, −10.26348173732390038409167161431, −8.001519252542740868927278602949, −6.92907628836750355063912450191, −6.01641053312472617821824930017, −4.84669149357568121655536873619, −3.64872431854918754860982818040, −1.31988523756809149834870520846, −0.937454972361249509658223623344,
0.937454972361249509658223623344, 1.31988523756809149834870520846, 3.64872431854918754860982818040, 4.84669149357568121655536873619, 6.01641053312472617821824930017, 6.92907628836750355063912450191, 8.001519252542740868927278602949, 10.26348173732390038409167161431, 11.17952798472451000381609481305, 11.85095090606809206981459199253
