| L(s) = 1 | + 3-s − 3.27·5-s + 9-s + 3.27·11-s − 6.27·13-s − 3.27·15-s + 4·17-s + 6.27·19-s + 4·23-s + 5.72·25-s + 27-s + 5.27·29-s + 31-s + 3.27·33-s − 2.27·37-s − 6.27·39-s + 4.54·41-s + 0.274·43-s − 3.27·45-s + 6·47-s + 4·51-s + 9.27·53-s − 10.7·55-s + 6.27·57-s + 1.27·59-s − 10·61-s + 20.5·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.46·5-s + 0.333·9-s + 0.987·11-s − 1.74·13-s − 0.845·15-s + 0.970·17-s + 1.43·19-s + 0.834·23-s + 1.14·25-s + 0.192·27-s + 0.979·29-s + 0.179·31-s + 0.570·33-s − 0.373·37-s − 1.00·39-s + 0.710·41-s + 0.0419·43-s − 0.488·45-s + 0.875·47-s + 0.560·51-s + 1.27·53-s − 1.44·55-s + 0.831·57-s + 0.165·59-s − 1.28·61-s + 2.54·65-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
| \(L(1)\) |
\(\approx\) |
\(1.584603557\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.584603557\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 3.27T + 5T^{2} \) |
| 11 | \( 1 - 3.27T + 11T^{2} \) |
| 13 | \( 1 + 6.27T + 13T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 - 6.27T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 - 5.27T + 29T^{2} \) |
| 31 | \( 1 - T + 31T^{2} \) |
| 37 | \( 1 + 2.27T + 37T^{2} \) |
| 41 | \( 1 - 4.54T + 41T^{2} \) |
| 43 | \( 1 - 0.274T + 43T^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 - 9.27T + 53T^{2} \) |
| 59 | \( 1 - 1.27T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 + 0.274T + 67T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 - 4.27T + 73T^{2} \) |
| 79 | \( 1 - 11.5T + 79T^{2} \) |
| 83 | \( 1 + 7.27T + 83T^{2} \) |
| 89 | \( 1 - 10.5T + 89T^{2} \) |
| 97 | \( 1 + 8.72T + 97T^{2} \) |
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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
−9.618864709079545015121161526342, −8.987495607621392752991652354507, −7.936453637277515419595218818133, −7.46086027889845111762886292274, −6.82630137123823443517049544064, −5.30685907922556240479389734562, −4.44125869578760822715080905559, −3.54877349360188194005071572453, −2.73216118317252566499188265594, −0.951447134868692497427977572661,
0.951447134868692497427977572661, 2.73216118317252566499188265594, 3.54877349360188194005071572453, 4.44125869578760822715080905559, 5.30685907922556240479389734562, 6.82630137123823443517049544064, 7.46086027889845111762886292274, 7.936453637277515419595218818133, 8.987495607621392752991652354507, 9.618864709079545015121161526342
