| L(s) = 1 | + 0.419·5-s − 4.23·7-s + 2.07·11-s − 5.80·13-s + 0.0751·17-s − 3.35·19-s − 4.32·23-s − 4.82·25-s − 8.99·29-s + 6.91·31-s − 1.77·35-s + 6.31·37-s − 4.28·41-s − 1.06·43-s + 9.61·47-s + 10.9·49-s + 14.0·53-s + 0.868·55-s − 3.20·59-s + 11.7·61-s − 2.43·65-s − 5.13·67-s + 0.458·71-s + 11.5·73-s − 8.77·77-s − 4.93·79-s + 9.02·83-s + ⋯ |
| L(s) = 1 | + 0.187·5-s − 1.59·7-s + 0.624·11-s − 1.61·13-s + 0.0182·17-s − 0.769·19-s − 0.901·23-s − 0.964·25-s − 1.66·29-s + 1.24·31-s − 0.299·35-s + 1.03·37-s − 0.669·41-s − 0.162·43-s + 1.40·47-s + 1.55·49-s + 1.93·53-s + 0.117·55-s − 0.417·59-s + 1.50·61-s − 0.301·65-s − 0.627·67-s + 0.0544·71-s + 1.34·73-s − 0.999·77-s − 0.554·79-s + 0.990·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9362087308\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9362087308\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 - 0.419T + 5T^{2} \) |
| 7 | \( 1 + 4.23T + 7T^{2} \) |
| 11 | \( 1 - 2.07T + 11T^{2} \) |
| 13 | \( 1 + 5.80T + 13T^{2} \) |
| 17 | \( 1 - 0.0751T + 17T^{2} \) |
| 19 | \( 1 + 3.35T + 19T^{2} \) |
| 23 | \( 1 + 4.32T + 23T^{2} \) |
| 29 | \( 1 + 8.99T + 29T^{2} \) |
| 31 | \( 1 - 6.91T + 31T^{2} \) |
| 37 | \( 1 - 6.31T + 37T^{2} \) |
| 41 | \( 1 + 4.28T + 41T^{2} \) |
| 43 | \( 1 + 1.06T + 43T^{2} \) |
| 47 | \( 1 - 9.61T + 47T^{2} \) |
| 53 | \( 1 - 14.0T + 53T^{2} \) |
| 59 | \( 1 + 3.20T + 59T^{2} \) |
| 61 | \( 1 - 11.7T + 61T^{2} \) |
| 67 | \( 1 + 5.13T + 67T^{2} \) |
| 71 | \( 1 - 0.458T + 71T^{2} \) |
| 73 | \( 1 - 11.5T + 73T^{2} \) |
| 79 | \( 1 + 4.93T + 79T^{2} \) |
| 83 | \( 1 - 9.02T + 83T^{2} \) |
| 89 | \( 1 + 13.3T + 89T^{2} \) |
| 97 | \( 1 - 10.7T + 97T^{2} \) |
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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
−7.983418157298721759744412591159, −7.26726743737343741997130260673, −6.64578774523163042275396334777, −6.02784386296428475809289072817, −5.37102356475124095533382736691, −4.21810343671115416858039656454, −3.76018152069489067313151695062, −2.67196917184876615173240175882, −2.08629490655831793258730578935, −0.47616921993962698163603373649,
0.47616921993962698163603373649, 2.08629490655831793258730578935, 2.67196917184876615173240175882, 3.76018152069489067313151695062, 4.21810343671115416858039656454, 5.37102356475124095533382736691, 6.02784386296428475809289072817, 6.64578774523163042275396334777, 7.26726743737343741997130260673, 7.983418157298721759744412591159
