| L(s) = 1 | + 1.56·3-s − 5-s + 7-s − 0.561·9-s − 5.56·11-s − 3.56·13-s − 1.56·15-s − 6.68·17-s − 4·19-s + 1.56·21-s + 25-s − 5.56·27-s + 0.438·29-s + 3.12·31-s − 8.68·33-s − 35-s + 9.12·37-s − 5.56·39-s + 8.24·41-s + 0.561·45-s − 2.43·47-s + 49-s − 10.4·51-s + 1.12·53-s + 5.56·55-s − 6.24·57-s + 2.24·59-s + ⋯ |
| L(s) = 1 | + 0.901·3-s − 0.447·5-s + 0.377·7-s − 0.187·9-s − 1.67·11-s − 0.987·13-s − 0.403·15-s − 1.62·17-s − 0.917·19-s + 0.340·21-s + 0.200·25-s − 1.07·27-s + 0.0814·29-s + 0.560·31-s − 1.51·33-s − 0.169·35-s + 1.49·37-s − 0.890·39-s + 1.28·41-s + 0.0837·45-s − 0.355·47-s + 0.142·49-s − 1.46·51-s + 0.154·53-s + 0.749·55-s − 0.827·57-s + 0.292·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 - 1.56T + 3T^{2} \) |
| 11 | \( 1 + 5.56T + 11T^{2} \) |
| 13 | \( 1 + 3.56T + 13T^{2} \) |
| 17 | \( 1 + 6.68T + 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 0.438T + 29T^{2} \) |
| 31 | \( 1 - 3.12T + 31T^{2} \) |
| 37 | \( 1 - 9.12T + 37T^{2} \) |
| 41 | \( 1 - 8.24T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 2.43T + 47T^{2} \) |
| 53 | \( 1 - 1.12T + 53T^{2} \) |
| 59 | \( 1 - 2.24T + 59T^{2} \) |
| 61 | \( 1 - 8.24T + 61T^{2} \) |
| 67 | \( 1 + 14.2T + 67T^{2} \) |
| 71 | \( 1 + 14.2T + 71T^{2} \) |
| 73 | \( 1 - 8.24T + 73T^{2} \) |
| 79 | \( 1 - 5.56T + 79T^{2} \) |
| 83 | \( 1 + 10.2T + 83T^{2} \) |
| 89 | \( 1 - 0.246T + 89T^{2} \) |
| 97 | \( 1 + 6.68T + 97T^{2} \) |
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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
−9.254152650460192690326561023032, −8.475377045757402409527477253906, −7.912278336088742263861396794187, −7.23380756133761694600234434498, −6.02907214214692386458706902232, −4.88879063482055374797909936040, −4.19266187088516058259925931270, −2.71636676424950522499192657456, −2.34406954505800109984968541948, 0,
2.34406954505800109984968541948, 2.71636676424950522499192657456, 4.19266187088516058259925931270, 4.88879063482055374797909936040, 6.02907214214692386458706902232, 7.23380756133761694600234434498, 7.912278336088742263861396794187, 8.475377045757402409527477253906, 9.254152650460192690326561023032
