| L(s) = 1 | − 5-s + 4·7-s − 3·9-s + 11-s − 2·13-s + 17-s − 2·19-s − 2·23-s + 25-s + 10·29-s − 4·35-s − 10·37-s + 4·41-s + 8·43-s + 3·45-s − 8·47-s + 9·49-s − 8·53-s − 55-s − 4·59-s − 2·61-s − 12·63-s + 2·65-s − 10·67-s − 12·71-s + 10·73-s + 4·77-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.51·7-s − 9-s + 0.301·11-s − 0.554·13-s + 0.242·17-s − 0.458·19-s − 0.417·23-s + 1/5·25-s + 1.85·29-s − 0.676·35-s − 1.64·37-s + 0.624·41-s + 1.21·43-s + 0.447·45-s − 1.16·47-s + 9/7·49-s − 1.09·53-s − 0.134·55-s − 0.520·59-s − 0.256·61-s − 1.51·63-s + 0.248·65-s − 1.22·67-s − 1.42·71-s + 1.17·73-s + 0.455·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.910014552\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.910014552\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−14.37462766033671, −14.00521430333824, −13.50520497222256, −12.56763061869800, −12.05586083582860, −11.97082936273009, −11.21286218458947, −10.88108058664820, −10.39681483848142, −9.710349678081137, −8.932997149484433, −8.617952212406782, −8.055879590758892, −7.731985416417902, −7.085692435448839, −6.346063773471184, −5.873075683079704, −5.051949754092547, −4.767320945381630, −4.211758912105733, −3.374318382027142, −2.780293181456842, −2.047262109192443, −1.388512034593583, −0.4785418763624650,
0.4785418763624650, 1.388512034593583, 2.047262109192443, 2.780293181456842, 3.374318382027142, 4.211758912105733, 4.767320945381630, 5.051949754092547, 5.873075683079704, 6.346063773471184, 7.085692435448839, 7.731985416417902, 8.055879590758892, 8.617952212406782, 8.932997149484433, 9.710349678081137, 10.39681483848142, 10.88108058664820, 11.21286218458947, 11.97082936273009, 12.05586083582860, 12.56763061869800, 13.50520497222256, 14.00521430333824, 14.37462766033671
