| L(s) = 1 | + 2.87·3-s + 5.26·9-s − 2.08·11-s + 0.695·13-s + 2.15·17-s + 8.21·19-s − 7.33·23-s + 6.51·27-s + 2.18·29-s + 7.88·31-s − 5.98·33-s − 1.78·37-s + 2·39-s − 1.48·41-s − 0.216·43-s + 11.6·47-s + 6.20·51-s − 3.14·53-s + 23.6·57-s − 12.4·59-s + 6.84·61-s − 13.1·67-s − 21.0·69-s + 10.3·71-s + 14.5·73-s + 10.9·79-s + 2.93·81-s + ⋯ |
| L(s) = 1 | + 1.65·3-s + 1.75·9-s − 0.627·11-s + 0.192·13-s + 0.523·17-s + 1.88·19-s − 1.52·23-s + 1.25·27-s + 0.405·29-s + 1.41·31-s − 1.04·33-s − 0.293·37-s + 0.320·39-s − 0.231·41-s − 0.0330·43-s + 1.70·47-s + 0.868·51-s − 0.432·53-s + 3.12·57-s − 1.62·59-s + 0.876·61-s − 1.60·67-s − 2.53·69-s + 1.22·71-s + 1.70·73-s + 1.22·79-s + 0.326·81-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
| \(L(1)\) |
\(\approx\) |
\(4.353644871\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.353644871\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.87T + 3T^{2} \) |
| 11 | \( 1 + 2.08T + 11T^{2} \) |
| 13 | \( 1 - 0.695T + 13T^{2} \) |
| 17 | \( 1 - 2.15T + 17T^{2} \) |
| 19 | \( 1 - 8.21T + 19T^{2} \) |
| 23 | \( 1 + 7.33T + 23T^{2} \) |
| 29 | \( 1 - 2.18T + 29T^{2} \) |
| 31 | \( 1 - 7.88T + 31T^{2} \) |
| 37 | \( 1 + 1.78T + 37T^{2} \) |
| 41 | \( 1 + 1.48T + 41T^{2} \) |
| 43 | \( 1 + 0.216T + 43T^{2} \) |
| 47 | \( 1 - 11.6T + 47T^{2} \) |
| 53 | \( 1 + 3.14T + 53T^{2} \) |
| 59 | \( 1 + 12.4T + 59T^{2} \) |
| 61 | \( 1 - 6.84T + 61T^{2} \) |
| 67 | \( 1 + 13.1T + 67T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 - 14.5T + 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 - 1.74T + 83T^{2} \) |
| 89 | \( 1 + 11.5T + 89T^{2} \) |
| 97 | \( 1 - 3.14T + 97T^{2} \) |
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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
−7.82783441527525961132937791143, −7.35042910227411261999548335033, −6.41391252864758791644038908985, −5.58572900851333170247485448698, −4.78907601973250923058162432086, −3.94115961214274598149366983327, −3.28464274126738600366794006477, −2.72409899527935005540046810409, −1.93695456670049236349803375876, −0.940200093283162898902412946913,
0.940200093283162898902412946913, 1.93695456670049236349803375876, 2.72409899527935005540046810409, 3.28464274126738600366794006477, 3.94115961214274598149366983327, 4.78907601973250923058162432086, 5.58572900851333170247485448698, 6.41391252864758791644038908985, 7.35042910227411261999548335033, 7.82783441527525961132937791143
