| L(s) = 1 | + 2.41·3-s − 5-s + 2.82·9-s − 11-s − 0.414·13-s − 2.41·15-s + 2.41·17-s + 2·19-s + 6.24·23-s + 25-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 − 2.82·45-s + 7.58·47-s + 5.82·51-s + 6.58·53-s + 55-s + 4.82·57-s + 1.75·59-s − 6.82·61-s + 0.414·65-s + ⋯ |
| L(s) = 1 | + 1.39·3-s − 0.447·5-s + 0.942·9-s − 0.301·11-s − 0.114·13-s − 0.623·15-s + 0.585·17-s + 0.458·19-s + 1.30·23-s + 0.200·25-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 − 0.421·45-s + 1.10·47-s + 0.816·51-s + 0.904·53-s + 0.134·55-s + 0.639·57-s + 0.228·59-s − 0.874·61-s + 0.0513·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.751843141\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.751843141\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 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
−9.053301150680697068463751732300, −8.378475245195067525704155718921, −7.75168032689530368302794288462, −7.16611213196274406822038201307, −6.07027822865347205374031962842, −4.93567605760188423545351986663, −4.09276614746034557592798850644, −3.06810725011219847628394458299, −2.61022829165235592374767122915, −1.10787533337244821661464067122,
1.10787533337244821661464067122, 2.61022829165235592374767122915, 3.06810725011219847628394458299, 4.09276614746034557592798850644, 4.93567605760188423545351986663, 6.07027822865347205374031962842, 7.16611213196274406822038201307, 7.75168032689530368302794288462, 8.378475245195067525704155718921, 9.053301150680697068463751732300
