| L(s) = 1 | − 2.41·2-s − 2·3-s + 3.82·4-s + 4.82·6-s + 2.82·7-s − 4.41·8-s + 9-s − 11-s − 7.65·12-s − 6.82·13-s − 6.82·14-s + 2.99·16-s + 4.82·17-s − 2.41·18-s − 19-s − 5.65·21-s + 2.41·22-s − 4·23-s + 8.82·24-s + 16.4·26-s + 4·27-s + 10.8·28-s + 3.17·29-s + 1.17·31-s + 1.58·32-s + 2·33-s − 11.6·34-s + ⋯ |
| L(s) = 1 | − 1.70·2-s − 1.15·3-s + 1.91·4-s + 1.97·6-s + 1.06·7-s − 1.56·8-s + 0.333·9-s − 0.301·11-s − 2.21·12-s − 1.89·13-s − 1.82·14-s + 0.749·16-s + 1.17·17-s − 0.569·18-s − 0.229·19-s − 1.23·21-s + 0.514·22-s − 0.834·23-s + 1.80·24-s + 3.23·26-s + 0.769·27-s + 2.04·28-s + 0.588·29-s + 0.210·31-s + 0.280·32-s + 0.348·33-s − 1.99·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + 2.41T + 2T^{2} \) |
| 3 | \( 1 + 2T + 3T^{2} \) |
| 7 | \( 1 - 2.82T + 7T^{2} \) |
| 13 | \( 1 + 6.82T + 13T^{2} \) |
| 17 | \( 1 - 4.82T + 17T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 3.17T + 29T^{2} \) |
| 31 | \( 1 - 1.17T + 31T^{2} \) |
| 37 | \( 1 - 1.17T + 37T^{2} \) |
| 41 | \( 1 + 8.82T + 41T^{2} \) |
| 43 | \( 1 - 6.82T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 2.82T + 53T^{2} \) |
| 59 | \( 1 - 12.4T + 59T^{2} \) |
| 61 | \( 1 + 3.65T + 61T^{2} \) |
| 67 | \( 1 + 0.343T + 67T^{2} \) |
| 71 | \( 1 - 9.17T + 71T^{2} \) |
| 73 | \( 1 - 10.4T + 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 + 2.82T + 83T^{2} \) |
| 89 | \( 1 + 13.3T + 89T^{2} \) |
| 97 | \( 1 + 10.1T + 97T^{2} \) |
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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
−8.101211789969037062980804442935, −7.23676322552597305833779700448, −6.75287869072939062097400271647, −5.70095390374492952867567242742, −5.16885079279985434694299406497, −4.39173508761125659924094710677, −2.78810026844554095991757947336, −1.97971899917903327055959651648, −0.956520665155033845358800161461, 0,
0.956520665155033845358800161461, 1.97971899917903327055959651648, 2.78810026844554095991757947336, 4.39173508761125659924094710677, 5.16885079279985434694299406497, 5.70095390374492952867567242742, 6.75287869072939062097400271647, 7.23676322552597305833779700448, 8.101211789969037062980804442935
