| L(s) = 1 | + 1.28·2-s − 0.338·4-s − 4.18·5-s − 3.67·7-s − 3.01·8-s − 5.39·10-s + 5.13·11-s + 4.32·13-s − 4.74·14-s − 3.20·16-s − 0.381·17-s + 1.30·19-s + 1.41·20-s + 6.62·22-s − 23-s + 12.5·25-s + 5.57·26-s + 1.24·28-s + 29-s + 5.12·31-s + 1.89·32-s − 0.491·34-s + 15.4·35-s − 6.52·37-s + 1.68·38-s + 12.6·40-s + 5.19·41-s + ⋯ |
| L(s) = 1 | + 0.911·2-s − 0.169·4-s − 1.87·5-s − 1.39·7-s − 1.06·8-s − 1.70·10-s + 1.54·11-s + 1.19·13-s − 1.26·14-s − 0.801·16-s − 0.0924·17-s + 0.300·19-s + 0.317·20-s + 1.41·22-s − 0.208·23-s + 2.50·25-s + 1.09·26-s + 0.235·28-s + 0.185·29-s + 0.920·31-s + 0.334·32-s − 0.0842·34-s + 2.60·35-s − 1.07·37-s + 0.273·38-s + 1.99·40-s + 0.811·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - 1.28T + 2T^{2} \) |
| 5 | \( 1 + 4.18T + 5T^{2} \) |
| 7 | \( 1 + 3.67T + 7T^{2} \) |
| 11 | \( 1 - 5.13T + 11T^{2} \) |
| 13 | \( 1 - 4.32T + 13T^{2} \) |
| 17 | \( 1 + 0.381T + 17T^{2} \) |
| 19 | \( 1 - 1.30T + 19T^{2} \) |
| 31 | \( 1 - 5.12T + 31T^{2} \) |
| 37 | \( 1 + 6.52T + 37T^{2} \) |
| 41 | \( 1 - 5.19T + 41T^{2} \) |
| 43 | \( 1 + 3.41T + 43T^{2} \) |
| 47 | \( 1 - 4.78T + 47T^{2} \) |
| 53 | \( 1 + 1.50T + 53T^{2} \) |
| 59 | \( 1 + 5.31T + 59T^{2} \) |
| 61 | \( 1 + 7.01T + 61T^{2} \) |
| 67 | \( 1 + 8.40T + 67T^{2} \) |
| 71 | \( 1 - 2.40T + 71T^{2} \) |
| 73 | \( 1 - 1.76T + 73T^{2} \) |
| 79 | \( 1 + 6.13T + 79T^{2} \) |
| 83 | \( 1 + 3.80T + 83T^{2} \) |
| 89 | \( 1 - 1.90T + 89T^{2} \) |
| 97 | \( 1 + 5.23T + 97T^{2} \) |
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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
−7.65546130996492150297654138187, −6.66153941325835933503291401409, −6.48592256016143434967467319446, −5.57056530666606524261111178058, −4.33882325550040898427027903855, −4.13665667722095777352305618339, −3.36382422570249986236653670249, −3.07483846001080213211696181213, −1.06403069994860018874825687648, 0,
1.06403069994860018874825687648, 3.07483846001080213211696181213, 3.36382422570249986236653670249, 4.13665667722095777352305618339, 4.33882325550040898427027903855, 5.57056530666606524261111178058, 6.48592256016143434967467319446, 6.66153941325835933503291401409, 7.65546130996492150297654138187
