| L(s) = 1 | − 3-s + 4-s + 9-s − 12-s + 13-s + 16-s − 19-s − 27-s + 2·31-s + 36-s + 37-s − 39-s − 2·43-s − 48-s + 52-s + 57-s − 61-s + 64-s + 67-s + 73-s − 76-s − 79-s + 81-s − 2·93-s + 97-s + 103-s − 108-s + ⋯ |
| L(s) = 1 | − 3-s + 4-s + 9-s − 12-s + 13-s + 16-s − 19-s − 27-s + 2·31-s + 36-s + 37-s − 39-s − 2·43-s − 48-s + 52-s + 57-s − 61-s + 64-s + 67-s + 73-s − 76-s − 79-s + 81-s − 2·93-s + 97-s + 103-s − 108-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.282204710\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.282204710\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( 1 - T + T^{2} \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( 1 + T + T^{2} \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 - T )^{2} \) |
| 37 | \( 1 - T + T^{2} \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 + T )^{2} \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( 1 - T + T^{2} \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( 1 - T + T^{2} \) |
| 79 | \( 1 + T + T^{2} \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( 1 - T + T^{2} \) |
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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
−8.499306569586201366138439303003, −7.938458253722916383846494265379, −6.97326937482390733319971681938, −6.35849549006967619998896852609, −6.03713792857122492235401455007, −5.01709101530938927610726815004, −4.19059701459342361384242528861, −3.20588321916529463906746644794, −2.07714338457109125094014824947, −1.08755182860160023856190543335,
1.08755182860160023856190543335, 2.07714338457109125094014824947, 3.20588321916529463906746644794, 4.19059701459342361384242528861, 5.01709101530938927610726815004, 6.03713792857122492235401455007, 6.35849549006967619998896852609, 6.97326937482390733319971681938, 7.938458253722916383846494265379, 8.499306569586201366138439303003
