| L(s) = 1 | + 1.00e3·2-s + 2.82e4·3-s + 7.39e5·4-s + 2.82e7·6-s + 4.78e8·8-s + 4.09e8·9-s + 2.08e10·12-s − 1.44e10·13-s + 2.84e11·16-s + 4.10e11·18-s − 1.80e12·23-s + 1.35e13·24-s + 3.81e12·25-s − 1.44e13·26-s + 6.30e11·27-s − 2.58e13·29-s + 4.63e13·31-s + 1.59e14·32-s + 3.03e14·36-s − 4.06e14·39-s − 5.38e14·41-s − 1.80e15·46-s − 2.01e15·47-s + 8.03e15·48-s + 1.62e15·49-s + 3.81e15·50-s − 1.06e16·52-s + ⋯ |
| L(s) = 1 | + 1.95·2-s + 1.43·3-s + 2.82·4-s + 2.80·6-s + 3.56·8-s + 1.05·9-s + 4.04·12-s − 1.35·13-s + 4.14·16-s + 2.06·18-s − 23-s + 5.11·24-s + 25-s − 2.65·26-s + 0.0826·27-s − 1.77·29-s + 1.75·31-s + 4.53·32-s + 2.98·36-s − 1.94·39-s − 1.64·41-s − 1.95·46-s − 1.80·47-s + 5.94·48-s + 49-s + 1.95·50-s − 3.83·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{19}{2})\) |
\(\approx\) |
\(11.67822918\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.67822918\) |
| \(L(10)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + p^{9} T \) |
| good | 2 | \( 1 - 1001 T + p^{18} T^{2} \) |
| 3 | \( 1 - 28234 T + p^{18} T^{2} \) |
| 5 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 7 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 11 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 13 | \( 1 + 14401098646 T + p^{18} T^{2} \) |
| 17 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 19 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 29 | \( 1 + 25800611881462 T + p^{18} T^{2} \) |
| 31 | \( 1 - 46387544624642 T + p^{18} T^{2} \) |
| 37 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 41 | \( 1 + 538722240191278 T + p^{18} T^{2} \) |
| 43 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 47 | \( 1 + 2018037720599134 T + p^{18} T^{2} \) |
| 53 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 59 | \( 1 - 15758386442415578 T + p^{18} T^{2} \) |
| 61 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 67 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 71 | \( 1 - 40558698720275762 T + p^{18} T^{2} \) |
| 73 | \( 1 - 51973347295686674 T + p^{18} T^{2} \) |
| 79 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 83 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 89 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 97 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
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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
−13.83877100077117935979267215193, −12.88298105588427695720233511801, −11.73368804452336289856193641480, −10.00654669708963870958569000323, −7.986220614631621341268842343611, −6.82422588773599657434278828448, −5.13038915151909024959381179601, −3.88305664283450861490644797012, −2.81275167606681373192316097020, −1.94329165165678285053919402493,
1.94329165165678285053919402493, 2.81275167606681373192316097020, 3.88305664283450861490644797012, 5.13038915151909024959381179601, 6.82422588773599657434278828448, 7.986220614631621341268842343611, 10.00654669708963870958569000323, 11.73368804452336289856193641480, 12.88298105588427695720233511801, 13.83877100077117935979267215193
