| L(s) = 1 | − 0.423·2-s + 3-s − 1.82·4-s + 3.41·5-s − 0.423·6-s + 1.61·8-s + 9-s − 1.44·10-s + 1.20·11-s − 1.82·12-s + 1.26·13-s + 3.41·15-s + 2.95·16-s + 2.17·17-s − 0.423·18-s − 7.30·19-s − 6.21·20-s − 0.512·22-s + 4.90·23-s + 1.61·24-s + 6.67·25-s − 0.535·26-s + 27-s + 0.506·29-s − 1.44·30-s + 6.04·31-s − 4.49·32-s + ⋯ |
| L(s) = 1 | − 0.299·2-s + 0.577·3-s − 0.910·4-s + 1.52·5-s − 0.173·6-s + 0.572·8-s + 0.333·9-s − 0.457·10-s + 0.364·11-s − 0.525·12-s + 0.350·13-s + 0.882·15-s + 0.738·16-s + 0.527·17-s − 0.0998·18-s − 1.67·19-s − 1.39·20-s − 0.109·22-s + 1.02·23-s + 0.330·24-s + 1.33·25-s − 0.104·26-s + 0.192·27-s + 0.0940·29-s − 0.264·30-s + 1.08·31-s − 0.793·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.589664815\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.589664815\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 + 0.423T + 2T^{2} \) |
| 5 | \( 1 - 3.41T + 5T^{2} \) |
| 11 | \( 1 - 1.20T + 11T^{2} \) |
| 13 | \( 1 - 1.26T + 13T^{2} \) |
| 17 | \( 1 - 2.17T + 17T^{2} \) |
| 19 | \( 1 + 7.30T + 19T^{2} \) |
| 23 | \( 1 - 4.90T + 23T^{2} \) |
| 29 | \( 1 - 0.506T + 29T^{2} \) |
| 31 | \( 1 - 6.04T + 31T^{2} \) |
| 37 | \( 1 - 6.73T + 37T^{2} \) |
| 43 | \( 1 + 7.21T + 43T^{2} \) |
| 47 | \( 1 - 5.13T + 47T^{2} \) |
| 53 | \( 1 + 7.67T + 53T^{2} \) |
| 59 | \( 1 - 3.05T + 59T^{2} \) |
| 61 | \( 1 - 1.50T + 61T^{2} \) |
| 67 | \( 1 - 2.66T + 67T^{2} \) |
| 71 | \( 1 + 8.67T + 71T^{2} \) |
| 73 | \( 1 - 12.5T + 73T^{2} \) |
| 79 | \( 1 - 7.94T + 79T^{2} \) |
| 83 | \( 1 + 5.53T + 83T^{2} \) |
| 89 | \( 1 - 4.64T + 89T^{2} \) |
| 97 | \( 1 + 1.13T + 97T^{2} \) |
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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
−8.497816303415050744081227684125, −7.50108032904379675723638895410, −6.50666568881810437507877387384, −6.07421148084746952688078012816, −5.11835411032261948578373965309, −4.52490261293803434765532752964, −3.62330258112557820578631726611, −2.65688764160992120243596322595, −1.77879070492520340023365676941, −0.924806902806545299701654307165,
0.924806902806545299701654307165, 1.77879070492520340023365676941, 2.65688764160992120243596322595, 3.62330258112557820578631726611, 4.52490261293803434765532752964, 5.11835411032261948578373965309, 6.07421148084746952688078012816, 6.50666568881810437507877387384, 7.50108032904379675723638895410, 8.497816303415050744081227684125
