| L(s) = 1 | − 3.11i·5-s − 7-s + 2.37i·11-s − 1.09i·13-s − 3.69·17-s + 1.08i·19-s + 4.87·23-s − 4.69·25-s − 1.59i·29-s − 7.45·31-s + 3.11i·35-s + 4.61i·37-s + 0.0380·41-s − 11.0i·43-s + 0.337·47-s + ⋯ |
| L(s) = 1 | − 1.39i·5-s − 0.377·7-s + 0.715i·11-s − 0.302i·13-s − 0.895·17-s + 0.248i·19-s + 1.01·23-s − 0.939·25-s − 0.296i·29-s − 1.33·31-s + 0.526i·35-s + 0.758i·37-s + 0.00594·41-s − 1.68i·43-s + 0.0491·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0430 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0430 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5792761787\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5792761787\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 + 3.11iT - 5T^{2} \) |
| 11 | \( 1 - 2.37iT - 11T^{2} \) |
| 13 | \( 1 + 1.09iT - 13T^{2} \) |
| 17 | \( 1 + 3.69T + 17T^{2} \) |
| 19 | \( 1 - 1.08iT - 19T^{2} \) |
| 23 | \( 1 - 4.87T + 23T^{2} \) |
| 29 | \( 1 + 1.59iT - 29T^{2} \) |
| 31 | \( 1 + 7.45T + 31T^{2} \) |
| 37 | \( 1 - 4.61iT - 37T^{2} \) |
| 41 | \( 1 - 0.0380T + 41T^{2} \) |
| 43 | \( 1 + 11.0iT - 43T^{2} \) |
| 47 | \( 1 - 0.337T + 47T^{2} \) |
| 53 | \( 1 - 8.14iT - 53T^{2} \) |
| 59 | \( 1 - 15.0iT - 59T^{2} \) |
| 61 | \( 1 + 5.53iT - 61T^{2} \) |
| 67 | \( 1 - 7.70iT - 67T^{2} \) |
| 71 | \( 1 + 13.5T + 71T^{2} \) |
| 73 | \( 1 + 14.6T + 73T^{2} \) |
| 79 | \( 1 + 5.18T + 79T^{2} \) |
| 83 | \( 1 + 0.138iT - 83T^{2} \) |
| 89 | \( 1 - 11.9T + 89T^{2} \) |
| 97 | \( 1 + 5.30T + 97T^{2} \) |
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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
−8.480439057657471296502004613309, −7.43963088859165527775519029824, −7.02978637784989102082602740782, −5.95316886242379181564899385613, −5.37303212634277361069981633953, −4.59407336050026934505849399247, −4.09099345139468052189400363407, −2.99744421414545494525699732766, −1.95965453744634703098334421335, −1.04332246970928010380644819562,
0.15657060296259254985112648467, 1.73167557124131294243240433345, 2.77181065279785400713172339292, 3.24187860342056595227488426671, 4.08054445836580895786996332474, 5.05734120652270746401177607860, 5.96531020196926784838025492619, 6.56332383449769391361556537556, 7.07114499660954711069579413806, 7.70187449559542391778265195046
