| L(s) = 1 | − 2.42·2-s + 3-s + 3.90·4-s − 2.42·6-s + 1.34·7-s − 4.61·8-s + 9-s + 3.90·12-s + 4.12·13-s − 3.25·14-s + 3.41·16-s − 5.87·17-s − 2.42·18-s − 3.51·19-s + 1.34·21-s + 6.37·23-s − 4.61·24-s − 10.0·26-s + 27-s + 5.23·28-s + 6.39·29-s + 9.40·31-s + 0.942·32-s + 14.2·34-s + 3.90·36-s + 7.01·37-s + 8.54·38-s + ⋯ |
| L(s) = 1 | − 1.71·2-s + 0.577·3-s + 1.95·4-s − 0.991·6-s + 0.507·7-s − 1.63·8-s + 0.333·9-s + 1.12·12-s + 1.14·13-s − 0.871·14-s + 0.853·16-s − 1.42·17-s − 0.572·18-s − 0.806·19-s + 0.292·21-s + 1.32·23-s − 0.942·24-s − 1.96·26-s + 0.192·27-s + 0.989·28-s + 1.18·29-s + 1.68·31-s + 0.166·32-s + 2.44·34-s + 0.650·36-s + 1.15·37-s + 1.38·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.304147537\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.304147537\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 2.42T + 2T^{2} \) |
| 7 | \( 1 - 1.34T + 7T^{2} \) |
| 13 | \( 1 - 4.12T + 13T^{2} \) |
| 17 | \( 1 + 5.87T + 17T^{2} \) |
| 19 | \( 1 + 3.51T + 19T^{2} \) |
| 23 | \( 1 - 6.37T + 23T^{2} \) |
| 29 | \( 1 - 6.39T + 29T^{2} \) |
| 31 | \( 1 - 9.40T + 31T^{2} \) |
| 37 | \( 1 - 7.01T + 37T^{2} \) |
| 41 | \( 1 + 8.63T + 41T^{2} \) |
| 43 | \( 1 - 5.31T + 43T^{2} \) |
| 47 | \( 1 - 3.74T + 47T^{2} \) |
| 53 | \( 1 + 6.41T + 53T^{2} \) |
| 59 | \( 1 - 8.97T + 59T^{2} \) |
| 61 | \( 1 + 6.52T + 61T^{2} \) |
| 67 | \( 1 - 8.83T + 67T^{2} \) |
| 71 | \( 1 + 7.17T + 71T^{2} \) |
| 73 | \( 1 + 4.08T + 73T^{2} \) |
| 79 | \( 1 - 9.97T + 79T^{2} \) |
| 83 | \( 1 - 3.06T + 83T^{2} \) |
| 89 | \( 1 + 11.8T + 89T^{2} \) |
| 97 | \( 1 + 2.55T + 97T^{2} \) |
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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
−8.211580562139476278406322828748, −7.21369581226649204447432147059, −6.60386831925605503579979229342, −6.19087813049700704933396357612, −4.81509691749450649102842268236, −4.23588983020023158336226469667, −3.01927124031469549646656055301, −2.38178911728198488919246666873, −1.50753485709923142149016523183, −0.74218313679778688738776506703,
0.74218313679778688738776506703, 1.50753485709923142149016523183, 2.38178911728198488919246666873, 3.01927124031469549646656055301, 4.23588983020023158336226469667, 4.81509691749450649102842268236, 6.19087813049700704933396357612, 6.60386831925605503579979229342, 7.21369581226649204447432147059, 8.211580562139476278406322828748
