| L(s) = 1 | − 3-s − 5-s + 7-s − 2·9-s + 11-s − 5·13-s + 15-s − 7·17-s − 4·19-s − 21-s − 23-s + 25-s + 5·27-s + 9·29-s − 6·31-s − 33-s − 35-s + 4·37-s + 5·39-s + 8·41-s − 2·43-s + 2·45-s + 5·47-s + 49-s + 7·51-s − 6·53-s − 55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.377·7-s − 2/3·9-s + 0.301·11-s − 1.38·13-s + 0.258·15-s − 1.69·17-s − 0.917·19-s − 0.218·21-s − 0.208·23-s + 1/5·25-s + 0.962·27-s + 1.67·29-s − 1.07·31-s − 0.174·33-s − 0.169·35-s + 0.657·37-s + 0.800·39-s + 1.24·41-s − 0.304·43-s + 0.298·45-s + 0.729·47-s + 1/7·49-s + 0.980·51-s − 0.824·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 5 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−14.61001049720386, −14.39041709862137, −13.84936789935926, −13.02800953699233, −12.64005737122029, −12.09824748815399, −11.70074151112758, −11.05329035511440, −10.87580721537001, −10.24330587008796, −9.463226999760150, −9.042520516344906, −8.380305213666909, −8.057051332569846, −7.219447986362982, −6.785303575463726, −6.285871016137006, −5.603696859292821, −5.000733070407736, −4.313926953484233, −4.221502433345187, −2.961958674540276, −2.517635144381789, −1.833143044937969, −0.6785436484141650, 0,
0.6785436484141650, 1.833143044937969, 2.517635144381789, 2.961958674540276, 4.221502433345187, 4.313926953484233, 5.000733070407736, 5.603696859292821, 6.285871016137006, 6.785303575463726, 7.219447986362982, 8.057051332569846, 8.380305213666909, 9.042520516344906, 9.463226999760150, 10.24330587008796, 10.87580721537001, 11.05329035511440, 11.70074151112758, 12.09824748815399, 12.64005737122029, 13.02800953699233, 13.84936789935926, 14.39041709862137, 14.61001049720386
