| L(s) = 1 | − 2.35·3-s − 2.56·5-s + 3.68·7-s + 2.56·9-s + 1.03·11-s + 6.04·15-s + 6.12·17-s − 5.00·19-s − 8.68·21-s − 7.07·23-s + 1.56·25-s + 1.03·27-s − 5·29-s − 3.39·31-s − 2.43·33-s − 9.43·35-s + 2.12·37-s − 4.12·41-s + 8.39·43-s − 6.56·45-s + 10.7·47-s + 6.56·49-s − 14.4·51-s − 2.56·53-s − 2.64·55-s + 11.8·57-s + 10.4·59-s + ⋯ |
| L(s) = 1 | − 1.36·3-s − 1.14·5-s + 1.39·7-s + 0.853·9-s + 0.311·11-s + 1.55·15-s + 1.48·17-s − 1.14·19-s − 1.89·21-s − 1.47·23-s + 0.312·25-s + 0.198·27-s − 0.928·29-s − 0.609·31-s − 0.424·33-s − 1.59·35-s + 0.349·37-s − 0.643·41-s + 1.28·43-s − 0.978·45-s + 1.56·47-s + 0.937·49-s − 2.02·51-s − 0.351·53-s − 0.357·55-s + 1.56·57-s + 1.36·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 2.35T + 3T^{2} \) |
| 5 | \( 1 + 2.56T + 5T^{2} \) |
| 7 | \( 1 - 3.68T + 7T^{2} \) |
| 11 | \( 1 - 1.03T + 11T^{2} \) |
| 17 | \( 1 - 6.12T + 17T^{2} \) |
| 19 | \( 1 + 5.00T + 19T^{2} \) |
| 23 | \( 1 + 7.07T + 23T^{2} \) |
| 29 | \( 1 + 5T + 29T^{2} \) |
| 31 | \( 1 + 3.39T + 31T^{2} \) |
| 37 | \( 1 - 2.12T + 37T^{2} \) |
| 41 | \( 1 + 4.12T + 41T^{2} \) |
| 43 | \( 1 - 8.39T + 43T^{2} \) |
| 47 | \( 1 - 10.7T + 47T^{2} \) |
| 53 | \( 1 + 2.56T + 53T^{2} \) |
| 59 | \( 1 - 10.4T + 59T^{2} \) |
| 61 | \( 1 + 11.2T + 61T^{2} \) |
| 67 | \( 1 + 9.72T + 67T^{2} \) |
| 71 | \( 1 - 5.00T + 71T^{2} \) |
| 73 | \( 1 + 4.31T + 73T^{2} \) |
| 79 | \( 1 - 11.5T + 79T^{2} \) |
| 83 | \( 1 + 2.64T + 83T^{2} \) |
| 89 | \( 1 - 10.6T + 89T^{2} \) |
| 97 | \( 1 + 9.80T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.70165139077318082870292786514, −7.29735628502787232372909012350, −6.13498485306699447728264424871, −5.70647603783661827129072809846, −4.89412465854300167719507547759, −4.22373322894030569008972495399, −3.65405601756521236856683092439, −2.11658832812393729024860836927, −1.09015417527562693985524363188, 0,
1.09015417527562693985524363188, 2.11658832812393729024860836927, 3.65405601756521236856683092439, 4.22373322894030569008972495399, 4.89412465854300167719507547759, 5.70647603783661827129072809846, 6.13498485306699447728264424871, 7.29735628502787232372909012350, 7.70165139077318082870292786514
