| L(s) = 1 | − 0.0716·3-s + 3.69·5-s + 1.67·7-s − 2.99·9-s + 5.28·11-s − 0.264·15-s − 0.533·17-s − 4.65·19-s − 0.119·21-s + 8.51·23-s + 8.65·25-s + 0.429·27-s + 4.48·29-s − 0.899·31-s − 0.378·33-s + 6.18·35-s − 3.17·37-s − 10.2·41-s − 5.24·43-s − 11.0·45-s + 4.09·47-s − 4.19·49-s + 0.0382·51-s − 9.76·53-s + 19.5·55-s + 0.333·57-s + 10.8·59-s + ⋯ |
| L(s) = 1 | − 0.0413·3-s + 1.65·5-s + 0.632·7-s − 0.998·9-s + 1.59·11-s − 0.0683·15-s − 0.129·17-s − 1.06·19-s − 0.0261·21-s + 1.77·23-s + 1.73·25-s + 0.0826·27-s + 0.832·29-s − 0.161·31-s − 0.0659·33-s + 1.04·35-s − 0.521·37-s − 1.59·41-s − 0.799·43-s − 1.64·45-s + 0.597·47-s − 0.599·49-s + 0.00535·51-s − 1.34·53-s + 2.63·55-s + 0.0441·57-s + 1.41·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.435133835\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.435133835\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 0.0716T + 3T^{2} \) |
| 5 | \( 1 - 3.69T + 5T^{2} \) |
| 7 | \( 1 - 1.67T + 7T^{2} \) |
| 11 | \( 1 - 5.28T + 11T^{2} \) |
| 17 | \( 1 + 0.533T + 17T^{2} \) |
| 19 | \( 1 + 4.65T + 19T^{2} \) |
| 23 | \( 1 - 8.51T + 23T^{2} \) |
| 29 | \( 1 - 4.48T + 29T^{2} \) |
| 31 | \( 1 + 0.899T + 31T^{2} \) |
| 37 | \( 1 + 3.17T + 37T^{2} \) |
| 41 | \( 1 + 10.2T + 41T^{2} \) |
| 43 | \( 1 + 5.24T + 43T^{2} \) |
| 47 | \( 1 - 4.09T + 47T^{2} \) |
| 53 | \( 1 + 9.76T + 53T^{2} \) |
| 59 | \( 1 - 10.8T + 59T^{2} \) |
| 61 | \( 1 - 0.994T + 61T^{2} \) |
| 67 | \( 1 + 11.7T + 67T^{2} \) |
| 71 | \( 1 - 12.3T + 71T^{2} \) |
| 73 | \( 1 - 4.70T + 73T^{2} \) |
| 79 | \( 1 - 8.31T + 79T^{2} \) |
| 83 | \( 1 - 12.5T + 83T^{2} \) |
| 89 | \( 1 + 2.12T + 89T^{2} \) |
| 97 | \( 1 + 13.2T + 97T^{2} \) |
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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
−9.429419895312575645851596205119, −8.912946678598808511480725899266, −8.302813252278167121628665900892, −6.66572222179075833500572058012, −6.51143158265823770946854474034, −5.40760668193443825084784495904, −4.77020113940522986569307450637, −3.36578908521484153922683183079, −2.21743896150152927739197748818, −1.30145696692293790667666483783,
1.30145696692293790667666483783, 2.21743896150152927739197748818, 3.36578908521484153922683183079, 4.77020113940522986569307450637, 5.40760668193443825084784495904, 6.51143158265823770946854474034, 6.66572222179075833500572058012, 8.302813252278167121628665900892, 8.912946678598808511480725899266, 9.429419895312575645851596205119
