| L(s) = 1 | − 2·2-s + 3-s + 2·4-s − 5-s − 2·6-s + 7-s + 9-s + 2·10-s + 2·11-s + 2·12-s − 2·14-s − 15-s − 4·16-s − 2·18-s − 19-s − 2·20-s + 21-s − 4·22-s + 3·23-s − 4·25-s + 27-s + 2·28-s − 5·29-s + 2·30-s − 9·31-s + 8·32-s + 2·33-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.577·3-s + 4-s − 0.447·5-s − 0.816·6-s + 0.377·7-s + 1/3·9-s + 0.632·10-s + 0.603·11-s + 0.577·12-s − 0.534·14-s − 0.258·15-s − 16-s − 0.471·18-s − 0.229·19-s − 0.447·20-s + 0.218·21-s − 0.852·22-s + 0.625·23-s − 4/5·25-s + 0.192·27-s + 0.377·28-s − 0.928·29-s + 0.365·30-s − 1.61·31-s + 1.41·32-s + 0.348·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 15 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - 17 T + p T^{2} \) |
| 97 | \( 1 + 3 T + p T^{2} \) |
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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
−8.225960645578745685481684887830, −7.67282947465342344141543122502, −7.16241120672809491152702346490, −6.28475334363675932470644079310, −5.10961451390126675263352515442, −4.17234345527637895256558824700, −3.38456274259336849941697333165, −2.10228652962663337954032846090, −1.39044544504831070602483731298, 0,
1.39044544504831070602483731298, 2.10228652962663337954032846090, 3.38456274259336849941697333165, 4.17234345527637895256558824700, 5.10961451390126675263352515442, 6.28475334363675932470644079310, 7.16241120672809491152702346490, 7.67282947465342344141543122502, 8.225960645578745685481684887830
