| L(s) = 1 | − 3.30·3-s + 0.148·5-s − 4.48·7-s + 7.92·9-s − 5.76·11-s + 2.52·13-s − 0.490·15-s − 1.39·17-s + 6.53·19-s + 14.8·21-s − 1.51·23-s − 4.97·25-s − 16.2·27-s − 7.16·29-s + 4.58·31-s + 19.0·33-s − 0.665·35-s − 3.08·37-s − 8.35·39-s + 1.44·41-s − 0.693·43-s + 1.17·45-s + 0.309·47-s + 13.1·49-s + 4.61·51-s + 9.64·53-s − 0.856·55-s + ⋯ |
| L(s) = 1 | − 1.90·3-s + 0.0663·5-s − 1.69·7-s + 2.64·9-s − 1.73·11-s + 0.701·13-s − 0.126·15-s − 0.338·17-s + 1.49·19-s + 3.23·21-s − 0.315·23-s − 0.995·25-s − 3.13·27-s − 1.32·29-s + 0.823·31-s + 3.31·33-s − 0.112·35-s − 0.506·37-s − 1.33·39-s + 0.225·41-s − 0.105·43-s + 0.175·45-s + 0.0451·47-s + 1.87·49-s + 0.646·51-s + 1.32·53-s − 0.115·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 751 | \( 1 + T \) |
| good | 3 | \( 1 + 3.30T + 3T^{2} \) |
| 5 | \( 1 - 0.148T + 5T^{2} \) |
| 7 | \( 1 + 4.48T + 7T^{2} \) |
| 11 | \( 1 + 5.76T + 11T^{2} \) |
| 13 | \( 1 - 2.52T + 13T^{2} \) |
| 17 | \( 1 + 1.39T + 17T^{2} \) |
| 19 | \( 1 - 6.53T + 19T^{2} \) |
| 23 | \( 1 + 1.51T + 23T^{2} \) |
| 29 | \( 1 + 7.16T + 29T^{2} \) |
| 31 | \( 1 - 4.58T + 31T^{2} \) |
| 37 | \( 1 + 3.08T + 37T^{2} \) |
| 41 | \( 1 - 1.44T + 41T^{2} \) |
| 43 | \( 1 + 0.693T + 43T^{2} \) |
| 47 | \( 1 - 0.309T + 47T^{2} \) |
| 53 | \( 1 - 9.64T + 53T^{2} \) |
| 59 | \( 1 - 14.8T + 59T^{2} \) |
| 61 | \( 1 - 14.2T + 61T^{2} \) |
| 67 | \( 1 + 6.92T + 67T^{2} \) |
| 71 | \( 1 - 7.49T + 71T^{2} \) |
| 73 | \( 1 - 1.54T + 73T^{2} \) |
| 79 | \( 1 + 9.02T + 79T^{2} \) |
| 83 | \( 1 + 0.305T + 83T^{2} \) |
| 89 | \( 1 + 8.31T + 89T^{2} \) |
| 97 | \( 1 + 14.7T + 97T^{2} \) |
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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
−7.28570015763328329327691218208, −7.00905921879213438052263735962, −6.06438605927390559719770354928, −5.64134031055791415716367042913, −5.23824903777558962655245303121, −4.11319203656644440930934599866, −3.39816074476015142390810633255, −2.25635320155073654030778338295, −0.814987989596458639881007945165, 0,
0.814987989596458639881007945165, 2.25635320155073654030778338295, 3.39816074476015142390810633255, 4.11319203656644440930934599866, 5.23824903777558962655245303121, 5.64134031055791415716367042913, 6.06438605927390559719770354928, 7.00905921879213438052263735962, 7.28570015763328329327691218208
