| L(s) = 1 | + 5-s + 4.38·7-s − 6.38·11-s − 1.13·13-s − 1.72·17-s + 19-s − 1.52·23-s + 25-s − 6.65·29-s − 3.72·31-s + 4.38·35-s − 4.59·37-s + 9.64·41-s + 2.59·43-s + 9.52·47-s + 12.2·49-s − 11.5·53-s − 6.38·55-s − 14.9·59-s − 1.79·61-s − 1.13·65-s − 7.04·67-s + 9.31·71-s − 8.50·73-s − 28.0·77-s − 9.25·79-s − 9.04·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.65·7-s − 1.92·11-s − 0.314·13-s − 0.419·17-s + 0.229·19-s − 0.317·23-s + 0.200·25-s − 1.23·29-s − 0.669·31-s + 0.741·35-s − 0.755·37-s + 1.50·41-s + 0.395·43-s + 1.38·47-s + 1.75·49-s − 1.58·53-s − 0.861·55-s − 1.95·59-s − 0.229·61-s − 0.140·65-s − 0.860·67-s + 1.10·71-s − 0.995·73-s − 3.19·77-s − 1.04·79-s − 0.992·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 - 4.38T + 7T^{2} \) |
| 11 | \( 1 + 6.38T + 11T^{2} \) |
| 13 | \( 1 + 1.13T + 13T^{2} \) |
| 17 | \( 1 + 1.72T + 17T^{2} \) |
| 23 | \( 1 + 1.52T + 23T^{2} \) |
| 29 | \( 1 + 6.65T + 29T^{2} \) |
| 31 | \( 1 + 3.72T + 31T^{2} \) |
| 37 | \( 1 + 4.59T + 37T^{2} \) |
| 41 | \( 1 - 9.64T + 41T^{2} \) |
| 43 | \( 1 - 2.59T + 43T^{2} \) |
| 47 | \( 1 - 9.52T + 47T^{2} \) |
| 53 | \( 1 + 11.5T + 53T^{2} \) |
| 59 | \( 1 + 14.9T + 59T^{2} \) |
| 61 | \( 1 + 1.79T + 61T^{2} \) |
| 67 | \( 1 + 7.04T + 67T^{2} \) |
| 71 | \( 1 - 9.31T + 71T^{2} \) |
| 73 | \( 1 + 8.50T + 73T^{2} \) |
| 79 | \( 1 + 9.25T + 79T^{2} \) |
| 83 | \( 1 + 9.04T + 83T^{2} \) |
| 89 | \( 1 + 10.5T + 89T^{2} \) |
| 97 | \( 1 + 4.59T + 97T^{2} \) |
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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
−7.58178813766866642362883252836, −7.28818001636418629776349474855, −5.89570525402236282959646016536, −5.50225968146912406581370478836, −4.82702315237388037218748704343, −4.22892718287735496638053139295, −2.93770090677344868696774201679, −2.21601320863620636397784602597, −1.50412561356942360633703288419, 0,
1.50412561356942360633703288419, 2.21601320863620636397784602597, 2.93770090677344868696774201679, 4.22892718287735496638053139295, 4.82702315237388037218748704343, 5.50225968146912406581370478836, 5.89570525402236282959646016536, 7.28818001636418629776349474855, 7.58178813766866642362883252836
