| L(s) = 1 | + 5-s + 2·7-s + 5·11-s − 5·13-s − 17-s + 2·19-s + 23-s + 25-s − 8·29-s + 7·31-s + 2·35-s − 4·37-s − 9·41-s − 43-s + 8·47-s − 3·49-s + 5·53-s + 5·55-s − 4·59-s + 4·61-s − 5·65-s − 11·67-s − 14·71-s − 8·73-s + 10·77-s − 8·79-s − 3·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.755·7-s + 1.50·11-s − 1.38·13-s − 0.242·17-s + 0.458·19-s + 0.208·23-s + 1/5·25-s − 1.48·29-s + 1.25·31-s + 0.338·35-s − 0.657·37-s − 1.40·41-s − 0.152·43-s + 1.16·47-s − 3/7·49-s + 0.686·53-s + 0.674·55-s − 0.520·59-s + 0.512·61-s − 0.620·65-s − 1.34·67-s − 1.66·71-s − 0.936·73-s + 1.13·77-s − 0.900·79-s − 0.329·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 9 T + p T^{2} \) |
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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
−15.11598879838003, −14.83540612994662, −14.35560843726026, −13.83320275471453, −13.39970573365380, −12.64567779704728, −12.03034582381919, −11.70104598342540, −11.28632212004571, −10.40281881450136, −10.01211232519928, −9.417098229836604, −8.880298376818562, −8.458744534697461, −7.504423937002004, −7.215199108815800, −6.587510955800386, −5.866290854013519, −5.290322937543518, −4.635639311348782, −4.163326858949108, −3.303838277019310, −2.555326888993921, −1.734852067622893, −1.262512496061641, 0,
1.262512496061641, 1.734852067622893, 2.555326888993921, 3.303838277019310, 4.163326858949108, 4.635639311348782, 5.290322937543518, 5.866290854013519, 6.587510955800386, 7.215199108815800, 7.504423937002004, 8.458744534697461, 8.880298376818562, 9.417098229836604, 10.01211232519928, 10.40281881450136, 11.28632212004571, 11.70104598342540, 12.03034582381919, 12.64567779704728, 13.39970573365380, 13.83320275471453, 14.35560843726026, 14.83540612994662, 15.11598879838003
