| L(s) = 1 | − 2.34·3-s + 5-s + 1.34·7-s + 2.50·9-s − 2.34·11-s + 1.50·13-s − 2.34·15-s − 5.19·17-s − 1.50·19-s − 3.15·21-s − 23-s + 25-s + 1.15·27-s + 8.85·29-s − 2.65·31-s + 5.50·33-s + 1.34·35-s + 11.5·37-s − 3.53·39-s − 5.89·41-s + 4·43-s + 2.50·45-s − 12.3·47-s − 5.18·49-s + 12.1·51-s + 7.84·53-s − 2.34·55-s + ⋯ |
| L(s) = 1 | − 1.35·3-s + 0.447·5-s + 0.508·7-s + 0.835·9-s − 0.707·11-s + 0.417·13-s − 0.605·15-s − 1.26·17-s − 0.345·19-s − 0.689·21-s − 0.208·23-s + 0.200·25-s + 0.223·27-s + 1.64·29-s − 0.476·31-s + 0.958·33-s + 0.227·35-s + 1.89·37-s − 0.565·39-s − 0.920·41-s + 0.609·43-s + 0.373·45-s − 1.80·47-s − 0.741·49-s + 1.70·51-s + 1.07·53-s − 0.316·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.044008289\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.044008289\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + 2.34T + 3T^{2} \) |
| 7 | \( 1 - 1.34T + 7T^{2} \) |
| 11 | \( 1 + 2.34T + 11T^{2} \) |
| 13 | \( 1 - 1.50T + 13T^{2} \) |
| 17 | \( 1 + 5.19T + 17T^{2} \) |
| 19 | \( 1 + 1.50T + 19T^{2} \) |
| 29 | \( 1 - 8.85T + 29T^{2} \) |
| 31 | \( 1 + 2.65T + 31T^{2} \) |
| 37 | \( 1 - 11.5T + 37T^{2} \) |
| 41 | \( 1 + 5.89T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 12.3T + 47T^{2} \) |
| 53 | \( 1 - 7.84T + 53T^{2} \) |
| 59 | \( 1 + 10.5T + 59T^{2} \) |
| 61 | \( 1 - 12.0T + 61T^{2} \) |
| 67 | \( 1 + 9.84T + 67T^{2} \) |
| 71 | \( 1 - 12.2T + 71T^{2} \) |
| 73 | \( 1 + 0.692T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 0.159T + 83T^{2} \) |
| 89 | \( 1 - 7.68T + 89T^{2} \) |
| 97 | \( 1 - 11.0T + 97T^{2} \) |
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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
−8.443956331261052416702500846484, −7.83405439170548594131247301250, −6.60216665030186922794469857021, −6.42794216033113550237698967893, −5.49767158267033251247739097082, −4.87156109678869337692858834663, −4.27205250326610831372463625089, −2.86149912523852990880460753249, −1.85124488882691768154566926698, −0.63627712865441610685739979158,
0.63627712865441610685739979158, 1.85124488882691768154566926698, 2.86149912523852990880460753249, 4.27205250326610831372463625089, 4.87156109678869337692858834663, 5.49767158267033251247739097082, 6.42794216033113550237698967893, 6.60216665030186922794469857021, 7.83405439170548594131247301250, 8.443956331261052416702500846484
