| L(s) = 1 | − 2·2-s − 3·3-s + 4·4-s − 5·5-s + 6·6-s − 8·8-s + 9·9-s + 10·10-s − 60·11-s − 12·12-s + 34·13-s + 15·15-s + 16·16-s − 42·17-s − 18·18-s + 76·19-s − 20·20-s + 120·22-s + 24·24-s + 25·25-s − 68·26-s − 27·27-s + 6·29-s − 30·30-s + 232·31-s − 32·32-s + 180·33-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 1.64·11-s − 0.288·12-s + 0.725·13-s + 0.258·15-s + 1/4·16-s − 0.599·17-s − 0.235·18-s + 0.917·19-s − 0.223·20-s + 1.16·22-s + 0.204·24-s + 1/5·25-s − 0.512·26-s − 0.192·27-s + 0.0384·29-s − 0.182·30-s + 1.34·31-s − 0.176·32-s + 0.949·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 3 | \( 1 + p T \) |
| 5 | \( 1 + p T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 60 T + p^{3} T^{2} \) |
| 13 | \( 1 - 34 T + p^{3} T^{2} \) |
| 17 | \( 1 + 42 T + p^{3} T^{2} \) |
| 19 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 23 | \( 1 + p^{3} T^{2} \) |
| 29 | \( 1 - 6 T + p^{3} T^{2} \) |
| 31 | \( 1 - 232 T + p^{3} T^{2} \) |
| 37 | \( 1 - 134 T + p^{3} T^{2} \) |
| 41 | \( 1 + 234 T + p^{3} T^{2} \) |
| 43 | \( 1 + 412 T + p^{3} T^{2} \) |
| 47 | \( 1 - 360 T + p^{3} T^{2} \) |
| 53 | \( 1 - 222 T + p^{3} T^{2} \) |
| 59 | \( 1 + 660 T + p^{3} T^{2} \) |
| 61 | \( 1 - 490 T + p^{3} T^{2} \) |
| 67 | \( 1 - 812 T + p^{3} T^{2} \) |
| 71 | \( 1 - 120 T + p^{3} T^{2} \) |
| 73 | \( 1 + 746 T + p^{3} T^{2} \) |
| 79 | \( 1 - 152 T + p^{3} T^{2} \) |
| 83 | \( 1 - 804 T + p^{3} T^{2} \) |
| 89 | \( 1 - 678 T + p^{3} T^{2} \) |
| 97 | \( 1 + 2 p T + p^{3} T^{2} \) |
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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
−8.566133886179145134429702390746, −8.020551828485846304147671354679, −7.23660053773631076587518781272, −6.39873574360615936342878832728, −5.46861604953167102930755625526, −4.67623525466600912610991708404, −3.40459617486998974543510152713, −2.38550360697533715404005323816, −1.00941817969652548790483069606, 0,
1.00941817969652548790483069606, 2.38550360697533715404005323816, 3.40459617486998974543510152713, 4.67623525466600912610991708404, 5.46861604953167102930755625526, 6.39873574360615936342878832728, 7.23660053773631076587518781272, 8.020551828485846304147671354679, 8.566133886179145134429702390746
