| L(s) = 1 | − 2.41·3-s + 2.82·9-s − 11-s + 0.414·13-s − 2.41·17-s + 2·19-s − 6.24·23-s + 0.414·27-s + 29-s + 10.2·31-s + 2.41·33-s − 11.8·37-s − 0.999·39-s + 4.58·41-s + 11.6·43-s − 7.58·47-s + 5.82·51-s − 6.58·53-s − 4.82·57-s + 1.75·59-s − 6.82·61-s + 1.41·67-s + 15.0·69-s − 2.48·71-s + 10.8·73-s − 3.34·79-s − 9.48·81-s + ⋯ |
| L(s) = 1 | − 1.39·3-s + 0.942·9-s − 0.301·11-s + 0.114·13-s − 0.585·17-s + 0.458·19-s − 1.30·23-s + 0.0797·27-s + 0.185·29-s + 1.83·31-s + 0.420·33-s − 1.95·37-s − 0.160·39-s + 0.716·41-s + 1.77·43-s − 1.10·47-s + 0.816·51-s − 0.904·53-s − 0.639·57-s + 0.228·59-s − 0.874·61-s + 0.172·67-s + 1.81·69-s − 0.294·71-s + 1.26·73-s − 0.376·79-s − 1.05·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 2.41T + 3T^{2} \) |
| 11 | \( 1 + T + 11T^{2} \) |
| 13 | \( 1 - 0.414T + 13T^{2} \) |
| 17 | \( 1 + 2.41T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 + 6.24T + 23T^{2} \) |
| 29 | \( 1 - T + 29T^{2} \) |
| 31 | \( 1 - 10.2T + 31T^{2} \) |
| 37 | \( 1 + 11.8T + 37T^{2} \) |
| 41 | \( 1 - 4.58T + 41T^{2} \) |
| 43 | \( 1 - 11.6T + 43T^{2} \) |
| 47 | \( 1 + 7.58T + 47T^{2} \) |
| 53 | \( 1 + 6.58T + 53T^{2} \) |
| 59 | \( 1 - 1.75T + 59T^{2} \) |
| 61 | \( 1 + 6.82T + 61T^{2} \) |
| 67 | \( 1 - 1.41T + 67T^{2} \) |
| 71 | \( 1 + 2.48T + 71T^{2} \) |
| 73 | \( 1 - 10.8T + 73T^{2} \) |
| 79 | \( 1 + 3.34T + 79T^{2} \) |
| 83 | \( 1 - 11.3T + 83T^{2} \) |
| 89 | \( 1 - 9.65T + 89T^{2} \) |
| 97 | \( 1 + 14.0T + 97T^{2} \) |
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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
−7.18779256961290935423428134968, −6.38004131890985047462811067508, −6.10945973186793186291165590726, −5.27410718495524675305981198899, −4.73988158462532262795917855376, −4.03531056985308966130380679726, −3.02491710069489775876042114060, −2.04029021116062404147737343646, −0.963746296292110783962708038889, 0,
0.963746296292110783962708038889, 2.04029021116062404147737343646, 3.02491710069489775876042114060, 4.03531056985308966130380679726, 4.73988158462532262795917855376, 5.27410718495524675305981198899, 6.10945973186793186291165590726, 6.38004131890985047462811067508, 7.18779256961290935423428134968
