| L(s) = 1 | − 3-s − 3.41·5-s + 9-s + 4.82·11-s − 1.41·13-s + 3.41·15-s + 6.24·17-s − 1.17·19-s − 0.828·23-s + 6.65·25-s − 27-s − 8.48·29-s − 10.8·31-s − 4.82·33-s − 9.65·37-s + 1.41·39-s + 3.41·41-s − 8·43-s − 3.41·45-s − 1.17·47-s − 6.24·51-s + 9.31·53-s − 16.4·55-s + 1.17·57-s − 10.8·59-s + 5.89·61-s + 4.82·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.52·5-s + 0.333·9-s + 1.45·11-s − 0.392·13-s + 0.881·15-s + 1.51·17-s − 0.268·19-s − 0.172·23-s + 1.33·25-s − 0.192·27-s − 1.57·29-s − 1.94·31-s − 0.840·33-s − 1.58·37-s + 0.226·39-s + 0.533·41-s − 1.21·43-s − 0.508·45-s − 0.170·47-s − 0.874·51-s + 1.27·53-s − 2.22·55-s + 0.155·57-s − 1.40·59-s + 0.755·61-s + 0.598·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 3.41T + 5T^{2} \) |
| 11 | \( 1 - 4.82T + 11T^{2} \) |
| 13 | \( 1 + 1.41T + 13T^{2} \) |
| 17 | \( 1 - 6.24T + 17T^{2} \) |
| 19 | \( 1 + 1.17T + 19T^{2} \) |
| 23 | \( 1 + 0.828T + 23T^{2} \) |
| 29 | \( 1 + 8.48T + 29T^{2} \) |
| 31 | \( 1 + 10.8T + 31T^{2} \) |
| 37 | \( 1 + 9.65T + 37T^{2} \) |
| 41 | \( 1 - 3.41T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 + 1.17T + 47T^{2} \) |
| 53 | \( 1 - 9.31T + 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 - 5.89T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 - 4.82T + 71T^{2} \) |
| 73 | \( 1 + 3.07T + 73T^{2} \) |
| 79 | \( 1 + 13.6T + 79T^{2} \) |
| 83 | \( 1 - 7.31T + 83T^{2} \) |
| 89 | \( 1 + 14.7T + 89T^{2} \) |
| 97 | \( 1 - 16.2T + 97T^{2} \) |
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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
−9.321555432034374677952770044323, −8.531280241852026142832813155333, −7.43352688800533565065674522014, −7.17236245321082160188805958132, −5.96161476003078248591631019613, −5.04567710423289581940539335574, −3.88346124659252457747948510552, −3.53488641975581685815856522423, −1.52637895715780074710635560934, 0,
1.52637895715780074710635560934, 3.53488641975581685815856522423, 3.88346124659252457747948510552, 5.04567710423289581940539335574, 5.96161476003078248591631019613, 7.17236245321082160188805958132, 7.43352688800533565065674522014, 8.531280241852026142832813155333, 9.321555432034374677952770044323
