| L(s) = 1 | + 2-s − 4-s − 2·5-s − 3·8-s − 2·10-s − 11-s + 13-s − 16-s − 4·19-s + 2·20-s − 22-s − 8·23-s − 25-s + 26-s − 10·29-s + 5·32-s − 6·37-s − 4·38-s + 6·40-s + 10·41-s + 4·43-s + 44-s − 8·46-s − 8·47-s − 7·49-s − 50-s − 52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.894·5-s − 1.06·8-s − 0.632·10-s − 0.301·11-s + 0.277·13-s − 1/4·16-s − 0.917·19-s + 0.447·20-s − 0.213·22-s − 1.66·23-s − 1/5·25-s + 0.196·26-s − 1.85·29-s + 0.883·32-s − 0.986·37-s − 0.648·38-s + 0.948·40-s + 1.56·41-s + 0.609·43-s + 0.150·44-s − 1.17·46-s − 1.16·47-s − 49-s − 0.141·50-s − 0.138·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 371943 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 371943 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−12.76115338170228, −12.35479835289522, −11.83722047160086, −11.53355930366592, −11.07426771355573, −10.47308718796249, −10.06369581520005, −9.532683560820502, −8.986774707281724, −8.664151542193970, −8.079574775589733, −7.728270417176014, −7.332381053112452, −6.576760654457231, −6.093042740169605, −5.681970603024907, −5.294978288076357, −4.484430732025391, −4.255458403289795, −3.778814642442659, −3.453201864516601, −2.731477660762958, −2.108348069783007, −1.511750630343075, −0.4442481384035594, 0,
0.4442481384035594, 1.511750630343075, 2.108348069783007, 2.731477660762958, 3.453201864516601, 3.778814642442659, 4.255458403289795, 4.484430732025391, 5.294978288076357, 5.681970603024907, 6.093042740169605, 6.576760654457231, 7.332381053112452, 7.728270417176014, 8.079574775589733, 8.664151542193970, 8.986774707281724, 9.532683560820502, 10.06369581520005, 10.47308718796249, 11.07426771355573, 11.53355930366592, 11.83722047160086, 12.35479835289522, 12.76115338170228
