| L(s) = 1 | − 1.41·3-s − 4.41·5-s − 7-s − 0.999·9-s − 4.24·11-s + 13-s + 6.24·15-s − 1.41·17-s + 1.24·19-s + 1.41·21-s + 0.171·23-s + 14.4·25-s + 5.65·27-s − 5.82·29-s + 5.24·31-s + 6·33-s + 4.41·35-s + 6.24·37-s − 1.41·39-s + 3.17·41-s − 5·43-s + 4.41·45-s − 4.41·47-s + 49-s + 2.00·51-s + 5.82·53-s + 18.7·55-s + ⋯ |
| L(s) = 1 | − 0.816·3-s − 1.97·5-s − 0.377·7-s − 0.333·9-s − 1.27·11-s + 0.277·13-s + 1.61·15-s − 0.342·17-s + 0.285·19-s + 0.308·21-s + 0.0357·23-s + 2.89·25-s + 1.08·27-s − 1.08·29-s + 0.941·31-s + 1.04·33-s + 0.746·35-s + 1.02·37-s − 0.226·39-s + 0.495·41-s − 0.762·43-s + 0.658·45-s − 0.643·47-s + 0.142·49-s + 0.280·51-s + 0.800·53-s + 2.52·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5824 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5824 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + 1.41T + 3T^{2} \) |
| 5 | \( 1 + 4.41T + 5T^{2} \) |
| 11 | \( 1 + 4.24T + 11T^{2} \) |
| 17 | \( 1 + 1.41T + 17T^{2} \) |
| 19 | \( 1 - 1.24T + 19T^{2} \) |
| 23 | \( 1 - 0.171T + 23T^{2} \) |
| 29 | \( 1 + 5.82T + 29T^{2} \) |
| 31 | \( 1 - 5.24T + 31T^{2} \) |
| 37 | \( 1 - 6.24T + 37T^{2} \) |
| 41 | \( 1 - 3.17T + 41T^{2} \) |
| 43 | \( 1 + 5T + 43T^{2} \) |
| 47 | \( 1 + 4.41T + 47T^{2} \) |
| 53 | \( 1 - 5.82T + 53T^{2} \) |
| 59 | \( 1 - 11.6T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 - 2.48T + 67T^{2} \) |
| 71 | \( 1 + 1.07T + 71T^{2} \) |
| 73 | \( 1 + 0.757T + 73T^{2} \) |
| 79 | \( 1 - 1.48T + 79T^{2} \) |
| 83 | \( 1 - 4.75T + 83T^{2} \) |
| 89 | \( 1 - 4.41T + 89T^{2} \) |
| 97 | \( 1 + 13.7T + 97T^{2} \) |
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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
−7.79997120587942786912909419426, −7.09902543140148337071017824720, −6.39360864685981151689778994116, −5.49594736932642507777461661547, −4.85832383909539669611700602068, −4.11932704155614598752210128739, −3.30770869602846898248048753742, −2.60078677154415704526410488448, −0.77455291951966662269228517423, 0,
0.77455291951966662269228517423, 2.60078677154415704526410488448, 3.30770869602846898248048753742, 4.11932704155614598752210128739, 4.85832383909539669611700602068, 5.49594736932642507777461661547, 6.39360864685981151689778994116, 7.09902543140148337071017824720, 7.79997120587942786912909419426
