| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s + 7-s − 8-s + 9-s + 10-s − 3·11-s + 12-s − 6·13-s − 14-s − 15-s + 16-s − 18-s + 4·19-s − 20-s + 21-s + 3·22-s − 24-s + 25-s + 6·26-s + 27-s + 28-s − 3·29-s + 30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.904·11-s + 0.288·12-s − 1.66·13-s − 0.267·14-s − 0.258·15-s + 1/4·16-s − 0.235·18-s + 0.917·19-s − 0.223·20-s + 0.218·21-s + 0.639·22-s − 0.204·24-s + 1/5·25-s + 1.17·26-s + 0.192·27-s + 0.188·28-s − 0.557·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 372810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 372810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 17 | \( 1 \) |
| 43 | \( 1 - T \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 11 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 - 7 T + p T^{2} \) |
| 89 | \( 1 + 5 T + p T^{2} \) |
| 97 | \( 1 + 19 T + p T^{2} \) |
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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
−12.72924406112397, −12.18626620742703, −11.82570456715168, −11.16991478284464, −11.03840363937699, −10.32837635077843, −9.887226547697375, −9.455109196676743, −9.334383266004715, −8.392604276785394, −8.216531511310484, −7.678413323177881, −7.396126838907351, −7.060878679647903, −6.394587607649609, −5.676190115736272, −5.185730206348317, −4.837142085818693, −4.153533058401655, −3.649623388776974, −2.906763265660720, −2.523412562777324, −2.185237091060296, −1.346696687124573, −0.6943898592671484, 0,
0.6943898592671484, 1.346696687124573, 2.185237091060296, 2.523412562777324, 2.906763265660720, 3.649623388776974, 4.153533058401655, 4.837142085818693, 5.185730206348317, 5.676190115736272, 6.394587607649609, 7.060878679647903, 7.396126838907351, 7.678413323177881, 8.216531511310484, 8.392604276785394, 9.334383266004715, 9.455109196676743, 9.887226547697375, 10.32837635077843, 11.03840363937699, 11.16991478284464, 11.82570456715168, 12.18626620742703, 12.72924406112397
