| L(s) = 1 | − 2-s + 3-s − 4-s − 6-s + 2·7-s + 3·8-s + 9-s − 12-s + 6·13-s − 2·14-s − 16-s − 4·17-s − 18-s − 2·19-s + 2·21-s − 23-s + 3·24-s − 5·25-s − 6·26-s + 27-s − 2·28-s − 2·29-s + 4·31-s − 5·32-s + 4·34-s − 36-s + 2·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.408·6-s + 0.755·7-s + 1.06·8-s + 1/3·9-s − 0.288·12-s + 1.66·13-s − 0.534·14-s − 1/4·16-s − 0.970·17-s − 0.235·18-s − 0.458·19-s + 0.436·21-s − 0.208·23-s + 0.612·24-s − 25-s − 1.17·26-s + 0.192·27-s − 0.377·28-s − 0.371·29-s + 0.718·31-s − 0.883·32-s + 0.685·34-s − 1/6·36-s + 0.328·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8349 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8349 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 11 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
| 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
−7.86008656096370547933386138145, −6.88943491307722467499715318162, −6.21871512819240029271961872665, −5.30418746340063878509114863866, −4.41304874455505109566961710379, −4.03740265174265937017429604240, −3.08949845024493465129587579827, −1.83112539889645102603939977123, −1.39507106301451678971086877698, 0,
1.39507106301451678971086877698, 1.83112539889645102603939977123, 3.08949845024493465129587579827, 4.03740265174265937017429604240, 4.41304874455505109566961710379, 5.30418746340063878509114863866, 6.21871512819240029271961872665, 6.88943491307722467499715318162, 7.86008656096370547933386138145
