| L(s) = 1 | − 1.41·3-s + 1.76·5-s − 0.999·9-s + 4.04·11-s − 1.13·13-s − 2.49·15-s − 4.43·17-s + 8.55·19-s − 5.96·23-s − 1.89·25-s + 5.65·27-s − 1.86·29-s − 5.60·31-s − 5.72·33-s − 1.95·37-s + 1.60·39-s − 8.97·41-s + 2.91·43-s − 1.76·45-s − 3.02·47-s + 6.27·51-s − 11.8·53-s + 7.14·55-s − 12.0·57-s + 6.94·59-s + 9.65·61-s − 2·65-s + ⋯ |
| L(s) = 1 | − 0.816·3-s + 0.788·5-s − 0.333·9-s + 1.22·11-s − 0.314·13-s − 0.643·15-s − 1.07·17-s + 1.96·19-s − 1.24·23-s − 0.378·25-s + 1.08·27-s − 0.345·29-s − 1.00·31-s − 0.996·33-s − 0.320·37-s + 0.256·39-s − 1.40·41-s + 0.444·43-s − 0.262·45-s − 0.440·47-s + 0.878·51-s − 1.63·53-s + 0.962·55-s − 1.60·57-s + 0.904·59-s + 1.23·61-s − 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5488 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 3 | \( 1 + 1.41T + 3T^{2} \) |
| 5 | \( 1 - 1.76T + 5T^{2} \) |
| 11 | \( 1 - 4.04T + 11T^{2} \) |
| 13 | \( 1 + 1.13T + 13T^{2} \) |
| 17 | \( 1 + 4.43T + 17T^{2} \) |
| 19 | \( 1 - 8.55T + 19T^{2} \) |
| 23 | \( 1 + 5.96T + 23T^{2} \) |
| 29 | \( 1 + 1.86T + 29T^{2} \) |
| 31 | \( 1 + 5.60T + 31T^{2} \) |
| 37 | \( 1 + 1.95T + 37T^{2} \) |
| 41 | \( 1 + 8.97T + 41T^{2} \) |
| 43 | \( 1 - 2.91T + 43T^{2} \) |
| 47 | \( 1 + 3.02T + 47T^{2} \) |
| 53 | \( 1 + 11.8T + 53T^{2} \) |
| 59 | \( 1 - 6.94T + 59T^{2} \) |
| 61 | \( 1 - 9.65T + 61T^{2} \) |
| 67 | \( 1 - 7.07T + 67T^{2} \) |
| 71 | \( 1 + 13.7T + 71T^{2} \) |
| 73 | \( 1 - 10.2T + 73T^{2} \) |
| 79 | \( 1 - 5.08T + 79T^{2} \) |
| 83 | \( 1 + 11.5T + 83T^{2} \) |
| 89 | \( 1 - 11.6T + 89T^{2} \) |
| 97 | \( 1 + 1.22T + 97T^{2} \) |
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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
−7.73091822572982923662114303812, −6.79651833864692213045227544196, −6.40811257352982084782001986967, −5.53452534554904718914602641819, −5.22255868666374386765061341370, −4.14155732168393308148207773927, −3.32682295307956853356896146554, −2.17943393802691787459516221821, −1.34291704815250087511166801879, 0,
1.34291704815250087511166801879, 2.17943393802691787459516221821, 3.32682295307956853356896146554, 4.14155732168393308148207773927, 5.22255868666374386765061341370, 5.53452534554904718914602641819, 6.40811257352982084782001986967, 6.79651833864692213045227544196, 7.73091822572982923662114303812
