| L(s) = 1 | − 3-s + 5-s + 7-s − 15-s + 2·17-s − 19-s − 21-s + 27-s + 29-s + 35-s − 41-s − 2·51-s + 53-s + 57-s + 59-s − 2·71-s + 79-s − 81-s + 2·85-s − 87-s − 95-s − 105-s − 107-s + 2·119-s + ⋯ |
| L(s) = 1 | − 3-s + 5-s + 7-s − 15-s + 2·17-s − 19-s − 21-s + 27-s + 29-s + 35-s − 41-s − 2·51-s + 53-s + 57-s + 59-s − 2·71-s + 79-s − 81-s + 2·85-s − 87-s − 95-s − 105-s − 107-s + 2·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.249730883\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.249730883\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 59 | \( 1 - T \) |
| good | 3 | \( 1 + T + T^{2} \) |
| 5 | \( 1 - T + T^{2} \) |
| 7 | \( 1 - T + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( ( 1 - T )^{2} \) |
| 19 | \( 1 + T + T^{2} \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( 1 - T + T^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( 1 - T + T^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 + T )^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( 1 - T + T^{2} \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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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
−8.545655935842705835282131189192, −8.060012764619162534142736137614, −7.06469131525232339365264380509, −6.25220601399780391395364569021, −5.62233964121146863442308610673, −5.19246454666977524065682687343, −4.35915826927891194613612272641, −3.12671240509553504743495259214, −2.01746087619403477112355416424, −1.08455899610548877700733893958,
1.08455899610548877700733893958, 2.01746087619403477112355416424, 3.12671240509553504743495259214, 4.35915826927891194613612272641, 5.19246454666977524065682687343, 5.62233964121146863442308610673, 6.25220601399780391395364569021, 7.06469131525232339365264380509, 8.060012764619162534142736137614, 8.545655935842705835282131189192
