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 − 4·11-s + 12·12-s − 62·13-s + 15·15-s + 16·16-s + 70·17-s − 18·18-s − 6·19-s + 20·20-s + 8·22-s − 70·23-s − 24·24-s + 25·25-s + 124·26-s + 27·27-s − 162·29-s − 30·30-s + 62·31-s − 32·32-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 − 0.109·11-s + 0.288·12-s − 1.32·13-s + 0.258·15-s + 1/4·16-s + 0.998·17-s − 0.235·18-s − 0.0724·19-s + 0.223·20-s + 0.0775·22-s − 0.634·23-s − 0.204·24-s + 1/5·25-s + 0.935·26-s + 0.192·27-s − 1.03·29-s − 0.182·30-s + 0.359·31-s − 0.176·32-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 | \( 1 + 4 T + p^{3} T^{2} \) |
| 13 | \( 1 + 62 T + p^{3} T^{2} \) |
| 17 | \( 1 - 70 T + p^{3} T^{2} \) |
| 19 | \( 1 + 6 T + p^{3} T^{2} \) |
| 23 | \( 1 + 70 T + p^{3} T^{2} \) |
| 29 | \( 1 + 162 T + p^{3} T^{2} \) |
| 31 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 37 | \( 1 - 218 T + p^{3} T^{2} \) |
| 41 | \( 1 + 130 T + p^{3} T^{2} \) |
| 43 | \( 1 - 232 T + p^{3} T^{2} \) |
| 47 | \( 1 + 304 T + p^{3} T^{2} \) |
| 53 | \( 1 + 380 T + p^{3} T^{2} \) |
| 59 | \( 1 + 376 T + p^{3} T^{2} \) |
| 61 | \( 1 + 56 T + p^{3} T^{2} \) |
| 67 | \( 1 + 952 T + p^{3} T^{2} \) |
| 71 | \( 1 - 708 T + p^{3} T^{2} \) |
| 73 | \( 1 + 682 T + p^{3} T^{2} \) |
| 79 | \( 1 + 8 p T + p^{3} T^{2} \) |
| 83 | \( 1 + 244 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1198 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1206 T + p^{3} T^{2} \) |
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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
−8.837875797701147820795841490931, −7.81639931960116291239324268575, −7.50315403402869801418419421861, −6.41796359911496694864807049398, −5.52935438489718241933270287371, −4.49611912157104462692204303665, −3.23255314705548289765276965495, −2.38294342113031334328345313575, −1.42409652796298005907600628567, 0,
1.42409652796298005907600628567, 2.38294342113031334328345313575, 3.23255314705548289765276965495, 4.49611912157104462692204303665, 5.52935438489718241933270287371, 6.41796359911496694864807049398, 7.50315403402869801418419421861, 7.81639931960116291239324268575, 8.837875797701147820795841490931