| L(s) = 1 | + 2·7-s − 3·9-s + 11-s + 6·13-s + 2·17-s + 4·19-s + 4·23-s − 6·29-s + 8·31-s − 8·37-s − 2·41-s + 10·43-s − 12·47-s − 3·49-s − 8·53-s + 2·61-s − 6·63-s − 4·67-s + 4·71-s + 10·73-s + 2·77-s + 16·79-s + 9·81-s − 6·83-s + 2·89-s + 12·91-s + 4·97-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 9-s + 0.301·11-s + 1.66·13-s + 0.485·17-s + 0.917·19-s + 0.834·23-s − 1.11·29-s + 1.43·31-s − 1.31·37-s − 0.312·41-s + 1.52·43-s − 1.75·47-s − 3/7·49-s − 1.09·53-s + 0.256·61-s − 0.755·63-s − 0.488·67-s + 0.474·71-s + 1.17·73-s + 0.227·77-s + 1.80·79-s + 81-s − 0.658·83-s + 0.211·89-s + 1.25·91-s + 0.406·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.303972426\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.303972426\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−8.251002943331823572490535645892, −7.902911684455703530936682843368, −6.83544638698109758613312984221, −6.11346953837921252845152007799, −5.42545842007195970604154867501, −4.75071631632763546008920236367, −3.60177411407712202518660280538, −3.11635487248471904600830410581, −1.80505071657773533984776239298, −0.907060218445421735956645133324,
0.907060218445421735956645133324, 1.80505071657773533984776239298, 3.11635487248471904600830410581, 3.60177411407712202518660280538, 4.75071631632763546008920236367, 5.42545842007195970604154867501, 6.11346953837921252845152007799, 6.83544638698109758613312984221, 7.902911684455703530936682843368, 8.251002943331823572490535645892
