| L(s) = 1 | − 0.363·3-s − 5-s − 1.14·7-s − 2.86·9-s + 2.72·11-s − 4.64·13-s + 0.363·15-s − 0.858·17-s + 19-s + 0.414·21-s + 4.41·23-s + 25-s + 2.13·27-s + 9.42·29-s + 10.2·31-s − 0.990·33-s + 1.14·35-s + 6.77·37-s + 1.68·39-s − 7.55·41-s − 9.29·43-s + 2.86·45-s + 7.00·47-s − 5.69·49-s + 0.311·51-s − 8.64·53-s − 2.72·55-s + ⋯ |
| L(s) = 1 | − 0.209·3-s − 0.447·5-s − 0.431·7-s − 0.955·9-s + 0.822·11-s − 1.28·13-s + 0.0938·15-s − 0.208·17-s + 0.229·19-s + 0.0904·21-s + 0.920·23-s + 0.200·25-s + 0.410·27-s + 1.74·29-s + 1.84·31-s − 0.172·33-s + 0.192·35-s + 1.11·37-s + 0.270·39-s − 1.18·41-s − 1.41·43-s + 0.427·45-s + 1.02·47-s − 0.813·49-s + 0.0436·51-s − 1.18·53-s − 0.367·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.143291632\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.143291632\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 0.363T + 3T^{2} \) |
| 7 | \( 1 + 1.14T + 7T^{2} \) |
| 11 | \( 1 - 2.72T + 11T^{2} \) |
| 13 | \( 1 + 4.64T + 13T^{2} \) |
| 17 | \( 1 + 0.858T + 17T^{2} \) |
| 23 | \( 1 - 4.41T + 23T^{2} \) |
| 29 | \( 1 - 9.42T + 29T^{2} \) |
| 31 | \( 1 - 10.2T + 31T^{2} \) |
| 37 | \( 1 - 6.77T + 37T^{2} \) |
| 41 | \( 1 + 7.55T + 41T^{2} \) |
| 43 | \( 1 + 9.29T + 43T^{2} \) |
| 47 | \( 1 - 7.00T + 47T^{2} \) |
| 53 | \( 1 + 8.64T + 53T^{2} \) |
| 59 | \( 1 - 5.14T + 59T^{2} \) |
| 61 | \( 1 - 9.45T + 61T^{2} \) |
| 67 | \( 1 - 13.6T + 67T^{2} \) |
| 71 | \( 1 - 5.45T + 71T^{2} \) |
| 73 | \( 1 - 6.87T + 73T^{2} \) |
| 79 | \( 1 + 17.2T + 79T^{2} \) |
| 83 | \( 1 - 14.2T + 83T^{2} \) |
| 89 | \( 1 - 13.0T + 89T^{2} \) |
| 97 | \( 1 + 9.68T + 97T^{2} \) |
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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
−9.556248985664498531486970664937, −8.571804636702404216864940377154, −8.019530542442729921635327995281, −6.76846619580337473901910700630, −6.49793365073257769129074264248, −5.19173351179998773516836582147, −4.56381929450625266197745502837, −3.31161230351380995223024530359, −2.54729615123918031800923671531, −0.75172710569529784104943551369,
0.75172710569529784104943551369, 2.54729615123918031800923671531, 3.31161230351380995223024530359, 4.56381929450625266197745502837, 5.19173351179998773516836582147, 6.49793365073257769129074264248, 6.76846619580337473901910700630, 8.019530542442729921635327995281, 8.571804636702404216864940377154, 9.556248985664498531486970664937
