| L(s) = 1 | − 3-s − 3.75·5-s − 7-s + 9-s − 4.90·11-s − 13-s + 3.75·15-s + 2.59·17-s + 6.06·19-s + 21-s − 0.249·23-s + 9.06·25-s − 27-s + 2.24·29-s + 8.34·31-s + 4.90·33-s + 3.75·35-s + 7.22·37-s + 39-s − 2.31·41-s − 4.24·43-s − 3.75·45-s − 6.84·47-s + 49-s − 2.59·51-s + 10.3·53-s + 18.4·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.67·5-s − 0.377·7-s + 0.333·9-s − 1.47·11-s − 0.277·13-s + 0.968·15-s + 0.628·17-s + 1.39·19-s + 0.218·21-s − 0.0520·23-s + 1.81·25-s − 0.192·27-s + 0.417·29-s + 1.49·31-s + 0.854·33-s + 0.633·35-s + 1.18·37-s + 0.160·39-s − 0.361·41-s − 0.648·43-s − 0.559·45-s − 0.998·47-s + 0.142·49-s − 0.363·51-s + 1.42·53-s + 2.48·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + 3.75T + 5T^{2} \) |
| 11 | \( 1 + 4.90T + 11T^{2} \) |
| 17 | \( 1 - 2.59T + 17T^{2} \) |
| 19 | \( 1 - 6.06T + 19T^{2} \) |
| 23 | \( 1 + 0.249T + 23T^{2} \) |
| 29 | \( 1 - 2.24T + 29T^{2} \) |
| 31 | \( 1 - 8.34T + 31T^{2} \) |
| 37 | \( 1 - 7.22T + 37T^{2} \) |
| 41 | \( 1 + 2.31T + 41T^{2} \) |
| 43 | \( 1 + 4.24T + 43T^{2} \) |
| 47 | \( 1 + 6.84T + 47T^{2} \) |
| 53 | \( 1 - 10.3T + 53T^{2} \) |
| 59 | \( 1 + 9.50T + 59T^{2} \) |
| 61 | \( 1 - 0.276T + 61T^{2} \) |
| 67 | \( 1 + 11.5T + 67T^{2} \) |
| 71 | \( 1 - 9.50T + 71T^{2} \) |
| 73 | \( 1 + 0.565T + 73T^{2} \) |
| 79 | \( 1 - 4.84T + 79T^{2} \) |
| 83 | \( 1 + 16.6T + 83T^{2} \) |
| 89 | \( 1 - 1.47T + 89T^{2} \) |
| 97 | \( 1 + 8.02T + 97T^{2} \) |
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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
−7.895721608108020092177033534240, −7.43000658688271067756274551636, −6.68701472740610973847610991722, −5.67819098959858167602103463452, −4.95803982845392147847388100947, −4.33881831713220123951715050349, −3.30494837503195802695343031298, −2.77329336162126322606210220410, −0.979978430424432740850167673230, 0,
0.979978430424432740850167673230, 2.77329336162126322606210220410, 3.30494837503195802695343031298, 4.33881831713220123951715050349, 4.95803982845392147847388100947, 5.67819098959858167602103463452, 6.68701472740610973847610991722, 7.43000658688271067756274551636, 7.895721608108020092177033534240
