| L(s) = 1 | + 3-s + 2.70·5-s + 4.70·7-s + 9-s − 4.70·11-s + 6·13-s + 2.70·15-s − 2.70·17-s − 19-s + 4.70·21-s − 4·23-s + 2.29·25-s + 27-s + 2·29-s − 9.40·31-s − 4.70·33-s + 12.7·35-s − 3.40·37-s + 6·39-s − 3.40·41-s − 10.1·43-s + 2.70·45-s + 0.701·47-s + 15.1·49-s − 2.70·51-s − 6·53-s − 12.7·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.20·5-s + 1.77·7-s + 0.333·9-s − 1.41·11-s + 1.66·13-s + 0.697·15-s − 0.655·17-s − 0.229·19-s + 1.02·21-s − 0.834·23-s + 0.459·25-s + 0.192·27-s + 0.371·29-s − 1.68·31-s − 0.818·33-s + 2.14·35-s − 0.559·37-s + 0.960·39-s − 0.531·41-s − 1.54·43-s + 0.402·45-s + 0.102·47-s + 2.15·49-s − 0.378·51-s − 0.824·53-s − 1.71·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.693794621\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.693794621\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 - 2.70T + 5T^{2} \) |
| 7 | \( 1 - 4.70T + 7T^{2} \) |
| 11 | \( 1 + 4.70T + 11T^{2} \) |
| 13 | \( 1 - 6T + 13T^{2} \) |
| 17 | \( 1 + 2.70T + 17T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 9.40T + 31T^{2} \) |
| 37 | \( 1 + 3.40T + 37T^{2} \) |
| 41 | \( 1 + 3.40T + 41T^{2} \) |
| 43 | \( 1 + 10.1T + 43T^{2} \) |
| 47 | \( 1 - 0.701T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 - 1.29T + 61T^{2} \) |
| 67 | \( 1 - 12T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 6.70T + 73T^{2} \) |
| 79 | \( 1 - 10.8T + 79T^{2} \) |
| 83 | \( 1 - 10.8T + 83T^{2} \) |
| 89 | \( 1 + 12.8T + 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−10.21588018878811312457583724357, −9.103661529193698508332495110919, −8.337972377884344595026549643463, −7.913501915863072375615715482465, −6.63898978128835554238771803913, −5.56611744304403174926503445464, −4.96893869047677703118976167912, −3.72251187072228488205090475641, −2.22050399435461860025427171118, −1.64209575666620308178801655306,
1.64209575666620308178801655306, 2.22050399435461860025427171118, 3.72251187072228488205090475641, 4.96893869047677703118976167912, 5.56611744304403174926503445464, 6.63898978128835554238771803913, 7.913501915863072375615715482465, 8.337972377884344595026549643463, 9.103661529193698508332495110919, 10.21588018878811312457583724357
