| L(s) = 1 | − 2.41·3-s + 3.82·5-s + 2.82·9-s + 0.414·11-s − 2.82·13-s − 9.24·15-s + 5.82·17-s − 3.58·19-s − 3.24·23-s + 9.65·25-s + 0.414·27-s − 2.82·29-s + 8.41·31-s − 0.999·33-s + 2.65·37-s + 6.82·39-s + 1.17·41-s − 1.65·43-s + 10.8·45-s + 7.58·47-s − 14.0·51-s + 53-s + 1.58·55-s + 8.65·57-s + 8.89·59-s − 2.65·61-s − 10.8·65-s + ⋯ |
| L(s) = 1 | − 1.39·3-s + 1.71·5-s + 0.942·9-s + 0.124·11-s − 0.784·13-s − 2.38·15-s + 1.41·17-s − 0.822·19-s − 0.676·23-s + 1.93·25-s + 0.0797·27-s − 0.525·29-s + 1.51·31-s − 0.174·33-s + 0.436·37-s + 1.09·39-s + 0.182·41-s − 0.252·43-s + 1.61·45-s + 1.10·47-s − 1.97·51-s + 0.137·53-s + 0.213·55-s + 1.14·57-s + 1.15·59-s − 0.340·61-s − 1.34·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.534600241\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.534600241\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 2.41T + 3T^{2} \) |
| 5 | \( 1 - 3.82T + 5T^{2} \) |
| 11 | \( 1 - 0.414T + 11T^{2} \) |
| 13 | \( 1 + 2.82T + 13T^{2} \) |
| 17 | \( 1 - 5.82T + 17T^{2} \) |
| 19 | \( 1 + 3.58T + 19T^{2} \) |
| 23 | \( 1 + 3.24T + 23T^{2} \) |
| 29 | \( 1 + 2.82T + 29T^{2} \) |
| 31 | \( 1 - 8.41T + 31T^{2} \) |
| 37 | \( 1 - 2.65T + 37T^{2} \) |
| 41 | \( 1 - 1.17T + 41T^{2} \) |
| 43 | \( 1 + 1.65T + 43T^{2} \) |
| 47 | \( 1 - 7.58T + 47T^{2} \) |
| 53 | \( 1 - T + 53T^{2} \) |
| 59 | \( 1 - 8.89T + 59T^{2} \) |
| 61 | \( 1 + 2.65T + 61T^{2} \) |
| 67 | \( 1 + 11.2T + 67T^{2} \) |
| 71 | \( 1 + 2.34T + 71T^{2} \) |
| 73 | \( 1 - 3.34T + 73T^{2} \) |
| 79 | \( 1 - 8.07T + 79T^{2} \) |
| 83 | \( 1 + 15.3T + 83T^{2} \) |
| 89 | \( 1 - 9T + 89T^{2} \) |
| 97 | \( 1 - 6.82T + 97T^{2} \) |
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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
−8.869884887397009584279657379376, −7.81750481234387884913088558762, −6.86864841439491532904611633135, −6.16951443675565341023424904593, −5.74327196550487352829081512160, −5.13955224829866439603582276058, −4.31855502182747054881020496535, −2.84807658647262013770843375560, −1.89355766794217234865307698622, −0.821670715045694906922317827463,
0.821670715045694906922317827463, 1.89355766794217234865307698622, 2.84807658647262013770843375560, 4.31855502182747054881020496535, 5.13955224829866439603582276058, 5.74327196550487352829081512160, 6.16951443675565341023424904593, 6.86864841439491532904611633135, 7.81750481234387884913088558762, 8.869884887397009584279657379376
