| L(s) = 1 | + 3-s + 1.41·5-s − 7-s + 9-s − 2·11-s + 2.82·13-s + 1.41·15-s − 4.24·17-s − 19-s − 21-s − 7.65·23-s − 2.99·25-s + 27-s + 3.41·29-s − 0.828·31-s − 2·33-s − 1.41·35-s − 6.48·37-s + 2.82·39-s + 9.65·41-s − 2·43-s + 1.41·45-s + 7.07·47-s + 49-s − 4.24·51-s − 7.89·53-s − 2.82·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.632·5-s − 0.377·7-s + 0.333·9-s − 0.603·11-s + 0.784·13-s + 0.365·15-s − 1.02·17-s − 0.229·19-s − 0.218·21-s − 1.59·23-s − 0.599·25-s + 0.192·27-s + 0.634·29-s − 0.148·31-s − 0.348·33-s − 0.239·35-s − 1.06·37-s + 0.452·39-s + 1.50·41-s − 0.304·43-s + 0.210·45-s + 1.03·47-s + 0.142·49-s − 0.594·51-s − 1.08·53-s − 0.381·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 - 1.41T + 5T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 2.82T + 13T^{2} \) |
| 17 | \( 1 + 4.24T + 17T^{2} \) |
| 23 | \( 1 + 7.65T + 23T^{2} \) |
| 29 | \( 1 - 3.41T + 29T^{2} \) |
| 31 | \( 1 + 0.828T + 31T^{2} \) |
| 37 | \( 1 + 6.48T + 37T^{2} \) |
| 41 | \( 1 - 9.65T + 41T^{2} \) |
| 43 | \( 1 + 2T + 43T^{2} \) |
| 47 | \( 1 - 7.07T + 47T^{2} \) |
| 53 | \( 1 + 7.89T + 53T^{2} \) |
| 59 | \( 1 + 2.34T + 59T^{2} \) |
| 61 | \( 1 + 4.82T + 61T^{2} \) |
| 67 | \( 1 + 12.4T + 67T^{2} \) |
| 71 | \( 1 + 12.5T + 71T^{2} \) |
| 73 | \( 1 + 0.343T + 73T^{2} \) |
| 79 | \( 1 + 14.8T + 79T^{2} \) |
| 83 | \( 1 - 1.41T + 83T^{2} \) |
| 89 | \( 1 + 4T + 89T^{2} \) |
| 97 | \( 1 - 7.65T + 97T^{2} \) |
| 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
−7.73457719929510219871589071620, −7.01638193396799772549840187107, −6.02728630424847501268027769975, −5.89733856407197796509835440553, −4.62568647253300143369020979669, −4.03147652774723404519188828377, −3.09114911291786305155162453603, −2.30401412091584285538803840811, −1.56529005693319799773220816051, 0,
1.56529005693319799773220816051, 2.30401412091584285538803840811, 3.09114911291786305155162453603, 4.03147652774723404519188828377, 4.62568647253300143369020979669, 5.89733856407197796509835440553, 6.02728630424847501268027769975, 7.01638193396799772549840187107, 7.73457719929510219871589071620
