| L(s) = 1 | + 0.656·2-s − 1.91·3-s − 1.56·4-s − 1.25·6-s − 2.56·7-s − 2.34·8-s + 0.656·9-s + 5.91·11-s + 3·12-s + 3.25·13-s − 1.68·14-s + 1.59·16-s + 5.31·17-s + 0.431·18-s + 4·19-s + 4.91·21-s + 3.88·22-s − 4·23-s + 4.48·24-s − 5·25-s + 2.13·26-s + 4.48·27-s + 4.03·28-s + 2.51·29-s − 9.13·31-s + 5.73·32-s − 11.3·33-s + ⋯ |
| L(s) = 1 | + 0.464·2-s − 1.10·3-s − 0.784·4-s − 0.512·6-s − 0.970·7-s − 0.828·8-s + 0.218·9-s + 1.78·11-s + 0.866·12-s + 0.902·13-s − 0.450·14-s + 0.399·16-s + 1.28·17-s + 0.101·18-s + 0.917·19-s + 1.07·21-s + 0.827·22-s − 0.834·23-s + 0.914·24-s − 25-s + 0.419·26-s + 0.862·27-s + 0.761·28-s + 0.466·29-s − 1.64·31-s + 1.01·32-s − 1.96·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9355334696\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9355334696\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 503 | \( 1 - T \) |
| good | 2 | \( 1 - 0.656T + 2T^{2} \) |
| 3 | \( 1 + 1.91T + 3T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 + 2.56T + 7T^{2} \) |
| 11 | \( 1 - 5.91T + 11T^{2} \) |
| 13 | \( 1 - 3.25T + 13T^{2} \) |
| 17 | \( 1 - 5.31T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 2.51T + 29T^{2} \) |
| 31 | \( 1 + 9.13T + 31T^{2} \) |
| 37 | \( 1 - 9.64T + 37T^{2} \) |
| 41 | \( 1 + 1.19T + 41T^{2} \) |
| 43 | \( 1 + 4.79T + 43T^{2} \) |
| 47 | \( 1 - 10.1T + 47T^{2} \) |
| 53 | \( 1 - 12.3T + 53T^{2} \) |
| 59 | \( 1 - 2.16T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 6.51T + 67T^{2} \) |
| 71 | \( 1 - 2.68T + 71T^{2} \) |
| 73 | \( 1 - 15.2T + 73T^{2} \) |
| 79 | \( 1 + 11.5T + 79T^{2} \) |
| 83 | \( 1 - 2.56T + 83T^{2} \) |
| 89 | \( 1 + 6.33T + 89T^{2} \) |
| 97 | \( 1 - 5.53T + 97T^{2} \) |
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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
−11.17124768674035092784799329867, −9.860620797470081357994513719235, −9.429197290545393277824965444233, −8.340288154042382636149119925324, −6.89802886609464085121091108866, −5.91246773657612240997375878283, −5.61200463532765327455218268820, −4.07755097682401091816323679767, −3.46295016204733397794952120390, −0.903566396325004342303843257014,
0.903566396325004342303843257014, 3.46295016204733397794952120390, 4.07755097682401091816323679767, 5.61200463532765327455218268820, 5.91246773657612240997375878283, 6.89802886609464085121091108866, 8.340288154042382636149119925324, 9.429197290545393277824965444233, 9.860620797470081357994513719235, 11.17124768674035092784799329867
