L(s) = 1 | − 2-s + 4-s + 5-s − 4·7-s − 8-s − 10-s + 4·11-s + 4·13-s + 4·14-s + 16-s − 4·17-s + 4·19-s + 20-s − 4·22-s − 8·23-s + 25-s − 4·26-s − 4·28-s − 6·29-s − 4·31-s − 32-s + 4·34-s − 4·35-s + 2·37-s − 4·38-s − 40-s − 10·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 1.51·7-s − 0.353·8-s − 0.316·10-s + 1.20·11-s + 1.10·13-s + 1.06·14-s + 1/4·16-s − 0.970·17-s + 0.917·19-s + 0.223·20-s − 0.852·22-s − 1.66·23-s + 1/5·25-s − 0.784·26-s − 0.755·28-s − 1.11·29-s − 0.718·31-s − 0.176·32-s + 0.685·34-s − 0.676·35-s + 0.328·37-s − 0.648·38-s − 0.158·40-s − 1.56·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3870 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3870 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 43 | \( 1 + T \) |
good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−8.345701922667043064965514705684, −7.26296647766519675085792894168, −6.54968858019624906103422844078, −6.22580923397546131317489562694, −5.40353800853593117429406911472, −3.84748900694382555903550940110, −3.56119506463623622222219749240, −2.30870965415115999411641018273, −1.35468288861878469024251602523, 0,
1.35468288861878469024251602523, 2.30870965415115999411641018273, 3.56119506463623622222219749240, 3.84748900694382555903550940110, 5.40353800853593117429406911472, 6.22580923397546131317489562694, 6.54968858019624906103422844078, 7.26296647766519675085792894168, 8.345701922667043064965514705684