| L(s) = 1 | + 2.53·2-s + 4.40·4-s + 2.62·7-s + 6.07·8-s + 4.12·11-s + 3.54·13-s + 6.65·14-s + 6.57·16-s − 3.91·17-s − 19-s + 10.4·22-s − 0.936·23-s + 8.97·26-s + 11.5·28-s − 1.01·29-s − 3.17·31-s + 4.48·32-s − 9.89·34-s + 7.54·37-s − 2.53·38-s − 1.95·41-s − 6.35·43-s + 18.1·44-s − 2.37·46-s + 8.97·47-s − 0.0840·49-s + 15.6·52-s + ⋯ |
| L(s) = 1 | + 1.78·2-s + 2.20·4-s + 0.993·7-s + 2.14·8-s + 1.24·11-s + 0.983·13-s + 1.77·14-s + 1.64·16-s − 0.948·17-s − 0.229·19-s + 2.22·22-s − 0.195·23-s + 1.75·26-s + 2.18·28-s − 0.189·29-s − 0.570·31-s + 0.793·32-s − 1.69·34-s + 1.24·37-s − 0.410·38-s − 0.305·41-s − 0.968·43-s + 2.73·44-s − 0.349·46-s + 1.30·47-s − 0.0120·49-s + 2.16·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.441244553\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.441244553\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - 2.53T + 2T^{2} \) |
| 7 | \( 1 - 2.62T + 7T^{2} \) |
| 11 | \( 1 - 4.12T + 11T^{2} \) |
| 13 | \( 1 - 3.54T + 13T^{2} \) |
| 17 | \( 1 + 3.91T + 17T^{2} \) |
| 23 | \( 1 + 0.936T + 23T^{2} \) |
| 29 | \( 1 + 1.01T + 29T^{2} \) |
| 31 | \( 1 + 3.17T + 31T^{2} \) |
| 37 | \( 1 - 7.54T + 37T^{2} \) |
| 41 | \( 1 + 1.95T + 41T^{2} \) |
| 43 | \( 1 + 6.35T + 43T^{2} \) |
| 47 | \( 1 - 8.97T + 47T^{2} \) |
| 53 | \( 1 + 11.1T + 53T^{2} \) |
| 59 | \( 1 - 7.01T + 59T^{2} \) |
| 61 | \( 1 - 12.6T + 61T^{2} \) |
| 67 | \( 1 + 3.98T + 67T^{2} \) |
| 71 | \( 1 + 7.01T + 71T^{2} \) |
| 73 | \( 1 - 7.16T + 73T^{2} \) |
| 79 | \( 1 + 11.5T + 79T^{2} \) |
| 83 | \( 1 + 14.2T + 83T^{2} \) |
| 89 | \( 1 + 13.1T + 89T^{2} \) |
| 97 | \( 1 - 3.54T + 97T^{2} \) |
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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
−8.320735118214828945153120385972, −7.30893143085063907228970155380, −6.65077015426099954119441652980, −6.04314162859593183330429108095, −5.35769273807770467502488675956, −4.39349555372846915253507374027, −4.12586607659532367697395731575, −3.23559488495787030824274219659, −2.13468265344418462524454691683, −1.38974150826902410798321559012,
1.38974150826902410798321559012, 2.13468265344418462524454691683, 3.23559488495787030824274219659, 4.12586607659532367697395731575, 4.39349555372846915253507374027, 5.35769273807770467502488675956, 6.04314162859593183330429108095, 6.65077015426099954119441652980, 7.30893143085063907228970155380, 8.320735118214828945153120385972
