| L(s) = 1 | + 0.414·2-s − 2·3-s − 1.82·4-s − 0.828·6-s − 2.82·7-s − 1.58·8-s + 9-s − 11-s + 3.65·12-s − 1.17·13-s − 1.17·14-s + 3·16-s − 0.828·17-s + 0.414·18-s − 19-s + 5.65·21-s − 0.414·22-s − 4·23-s + 3.17·24-s − 0.485·26-s + 4·27-s + 5.17·28-s + 8.82·29-s + 6.82·31-s + 4.41·32-s + 2·33-s − 0.343·34-s + ⋯ |
| L(s) = 1 | + 0.292·2-s − 1.15·3-s − 0.914·4-s − 0.338·6-s − 1.06·7-s − 0.560·8-s + 0.333·9-s − 0.301·11-s + 1.05·12-s − 0.324·13-s − 0.313·14-s + 0.750·16-s − 0.200·17-s + 0.0976·18-s − 0.229·19-s + 1.23·21-s − 0.0883·22-s − 0.834·23-s + 0.647·24-s − 0.0951·26-s + 0.769·27-s + 0.977·28-s + 1.63·29-s + 1.22·31-s + 0.780·32-s + 0.348·33-s − 0.0588·34-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 | \( 1 - 0.414T + 2T^{2} \) |
| 3 | \( 1 + 2T + 3T^{2} \) |
| 7 | \( 1 + 2.82T + 7T^{2} \) |
| 13 | \( 1 + 1.17T + 13T^{2} \) |
| 17 | \( 1 + 0.828T + 17T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 8.82T + 29T^{2} \) |
| 31 | \( 1 - 6.82T + 31T^{2} \) |
| 37 | \( 1 - 6.82T + 37T^{2} \) |
| 41 | \( 1 + 3.17T + 41T^{2} \) |
| 43 | \( 1 - 1.17T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 2.82T + 53T^{2} \) |
| 59 | \( 1 + 4.48T + 59T^{2} \) |
| 61 | \( 1 - 7.65T + 61T^{2} \) |
| 67 | \( 1 + 11.6T + 67T^{2} \) |
| 71 | \( 1 - 14.8T + 71T^{2} \) |
| 73 | \( 1 + 6.48T + 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 - 2.82T + 83T^{2} \) |
| 89 | \( 1 - 9.31T + 89T^{2} \) |
| 97 | \( 1 - 18.1T + 97T^{2} \) |
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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
−7.915627031940156486744879197016, −6.78874063464283840126482280853, −6.21049630499028203254347388487, −5.78270421160265564611191014598, −4.80625043022799455195644340897, −4.45805381325318463365338623450, −3.37162516470681093340324860019, −2.59703432274511388272124605615, −0.863794095961145059042650541089, 0,
0.863794095961145059042650541089, 2.59703432274511388272124605615, 3.37162516470681093340324860019, 4.45805381325318463365338623450, 4.80625043022799455195644340897, 5.78270421160265564611191014598, 6.21049630499028203254347388487, 6.78874063464283840126482280853, 7.915627031940156486744879197016
