| L(s) = 1 | − 2-s − 1.03·3-s + 4-s + 2.52·5-s + 1.03·6-s − 1.73·7-s − 8-s − 1.92·9-s − 2.52·10-s + 11-s − 1.03·12-s − 0.661·13-s + 1.73·14-s − 2.61·15-s + 16-s − 2.09·17-s + 1.92·18-s + 0.592·19-s + 2.52·20-s + 1.79·21-s − 22-s − 1.91·23-s + 1.03·24-s + 1.35·25-s + 0.661·26-s + 5.10·27-s − 1.73·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.597·3-s + 0.5·4-s + 1.12·5-s + 0.422·6-s − 0.656·7-s − 0.353·8-s − 0.642·9-s − 0.797·10-s + 0.301·11-s − 0.298·12-s − 0.183·13-s + 0.463·14-s − 0.674·15-s + 0.250·16-s − 0.508·17-s + 0.454·18-s + 0.135·19-s + 0.563·20-s + 0.392·21-s − 0.213·22-s − 0.399·23-s + 0.211·24-s + 0.271·25-s + 0.129·26-s + 0.982·27-s − 0.328·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 197 | \( 1 - T \) |
| good | 3 | \( 1 + 1.03T + 3T^{2} \) |
| 5 | \( 1 - 2.52T + 5T^{2} \) |
| 7 | \( 1 + 1.73T + 7T^{2} \) |
| 13 | \( 1 + 0.661T + 13T^{2} \) |
| 17 | \( 1 + 2.09T + 17T^{2} \) |
| 19 | \( 1 - 0.592T + 19T^{2} \) |
| 23 | \( 1 + 1.91T + 23T^{2} \) |
| 29 | \( 1 - 5.77T + 29T^{2} \) |
| 31 | \( 1 - 4.37T + 31T^{2} \) |
| 37 | \( 1 + 7.16T + 37T^{2} \) |
| 41 | \( 1 + 1.33T + 41T^{2} \) |
| 43 | \( 1 - 9.79T + 43T^{2} \) |
| 47 | \( 1 - 1.79T + 47T^{2} \) |
| 53 | \( 1 - 9.08T + 53T^{2} \) |
| 59 | \( 1 + 9.26T + 59T^{2} \) |
| 61 | \( 1 + 2.08T + 61T^{2} \) |
| 67 | \( 1 - 0.726T + 67T^{2} \) |
| 71 | \( 1 + 7.98T + 71T^{2} \) |
| 73 | \( 1 - 11.4T + 73T^{2} \) |
| 79 | \( 1 + 14.6T + 79T^{2} \) |
| 83 | \( 1 + 3.52T + 83T^{2} \) |
| 89 | \( 1 + 13.6T + 89T^{2} \) |
| 97 | \( 1 + 3.68T + 97T^{2} \) |
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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
−8.203580143499463027601573769375, −7.09080960505476516987212301970, −6.50187433045674736883714853773, −5.94853293618701474132078826582, −5.36533004057535856189029801553, −4.31026605323313157082460465286, −3.03584702690233244087659767616, −2.36629118597613319846338845450, −1.24671740740159360699391844819, 0,
1.24671740740159360699391844819, 2.36629118597613319846338845450, 3.03584702690233244087659767616, 4.31026605323313157082460465286, 5.36533004057535856189029801553, 5.94853293618701474132078826582, 6.50187433045674736883714853773, 7.09080960505476516987212301970, 8.203580143499463027601573769375
