| L(s) = 1 | + 1.60·2-s − 2.82·3-s + 0.574·4-s − 2.43·5-s − 4.53·6-s − 2.28·8-s + 4.99·9-s − 3.90·10-s + 1.78·11-s − 1.62·12-s + 3.58·13-s + 6.89·15-s − 4.81·16-s + 3.06·17-s + 8.02·18-s − 8.27·19-s − 1.39·20-s + 2.86·22-s + 2.42·23-s + 6.47·24-s + 0.938·25-s + 5.74·26-s − 5.65·27-s + 8.30·29-s + 11.0·30-s − 1.71·31-s − 3.15·32-s + ⋯ |
| L(s) = 1 | + 1.13·2-s − 1.63·3-s + 0.287·4-s − 1.08·5-s − 1.85·6-s − 0.808·8-s + 1.66·9-s − 1.23·10-s + 0.538·11-s − 0.468·12-s + 0.993·13-s + 1.77·15-s − 1.20·16-s + 0.743·17-s + 1.89·18-s − 1.89·19-s − 0.312·20-s + 0.611·22-s + 0.505·23-s + 1.32·24-s + 0.187·25-s + 1.12·26-s − 1.08·27-s + 1.54·29-s + 2.01·30-s − 0.307·31-s − 0.557·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3577 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3577 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 73 | \( 1 - T \) |
| good | 2 | \( 1 - 1.60T + 2T^{2} \) |
| 3 | \( 1 + 2.82T + 3T^{2} \) |
| 5 | \( 1 + 2.43T + 5T^{2} \) |
| 11 | \( 1 - 1.78T + 11T^{2} \) |
| 13 | \( 1 - 3.58T + 13T^{2} \) |
| 17 | \( 1 - 3.06T + 17T^{2} \) |
| 19 | \( 1 + 8.27T + 19T^{2} \) |
| 23 | \( 1 - 2.42T + 23T^{2} \) |
| 29 | \( 1 - 8.30T + 29T^{2} \) |
| 31 | \( 1 + 1.71T + 31T^{2} \) |
| 37 | \( 1 - 8.32T + 37T^{2} \) |
| 41 | \( 1 - 9.15T + 41T^{2} \) |
| 43 | \( 1 - 0.763T + 43T^{2} \) |
| 47 | \( 1 + 8.97T + 47T^{2} \) |
| 53 | \( 1 + 0.940T + 53T^{2} \) |
| 59 | \( 1 + 3.40T + 59T^{2} \) |
| 61 | \( 1 - 4.81T + 61T^{2} \) |
| 67 | \( 1 + 7.17T + 67T^{2} \) |
| 71 | \( 1 + 11.5T + 71T^{2} \) |
| 79 | \( 1 + 7.81T + 79T^{2} \) |
| 83 | \( 1 - 2.50T + 83T^{2} \) |
| 89 | \( 1 + 13.8T + 89T^{2} \) |
| 97 | \( 1 + 11.5T + 97T^{2} \) |
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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
−8.108694611219156029942793430418, −7.05736678706477862047595618350, −6.21954099511579833301686606234, −6.07488330320502413890417745824, −5.04699592625283940238645538266, −4.25713465003702679288852037108, −4.06140857708508084015495727613, −2.90663328802625543352902423373, −1.14360380755796734707707102760, 0,
1.14360380755796734707707102760, 2.90663328802625543352902423373, 4.06140857708508084015495727613, 4.25713465003702679288852037108, 5.04699592625283940238645538266, 6.07488330320502413890417745824, 6.21954099511579833301686606234, 7.05736678706477862047595618350, 8.108694611219156029942793430418
