| L(s) = 1 | + 2.15·3-s + 1.11·5-s − 1.03·7-s + 1.65·9-s + 2.85·11-s + 3.74·13-s + 2.40·15-s + 5.19·17-s + 1.79·19-s − 2.23·21-s + 6.06·23-s − 3.75·25-s − 2.90·27-s + 0.462·29-s + 1.21·31-s + 6.14·33-s − 1.15·35-s − 7.34·37-s + 8.06·39-s + 6.52·41-s + 2.05·43-s + 1.84·45-s − 5.41·47-s − 5.92·49-s + 11.2·51-s + 2.63·53-s + 3.18·55-s + ⋯ |
| L(s) = 1 | + 1.24·3-s + 0.499·5-s − 0.391·7-s + 0.550·9-s + 0.859·11-s + 1.03·13-s + 0.621·15-s + 1.26·17-s + 0.411·19-s − 0.487·21-s + 1.26·23-s − 0.750·25-s − 0.559·27-s + 0.0859·29-s + 0.217·31-s + 1.07·33-s − 0.195·35-s − 1.20·37-s + 1.29·39-s + 1.01·41-s + 0.314·43-s + 0.274·45-s − 0.790·47-s − 0.846·49-s + 1.56·51-s + 0.362·53-s + 0.429·55-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
| \(L(1)\) |
\(\approx\) |
\(4.009624376\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.009624376\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.15T + 3T^{2} \) |
| 5 | \( 1 - 1.11T + 5T^{2} \) |
| 7 | \( 1 + 1.03T + 7T^{2} \) |
| 11 | \( 1 - 2.85T + 11T^{2} \) |
| 13 | \( 1 - 3.74T + 13T^{2} \) |
| 17 | \( 1 - 5.19T + 17T^{2} \) |
| 19 | \( 1 - 1.79T + 19T^{2} \) |
| 23 | \( 1 - 6.06T + 23T^{2} \) |
| 29 | \( 1 - 0.462T + 29T^{2} \) |
| 31 | \( 1 - 1.21T + 31T^{2} \) |
| 37 | \( 1 + 7.34T + 37T^{2} \) |
| 41 | \( 1 - 6.52T + 41T^{2} \) |
| 43 | \( 1 - 2.05T + 43T^{2} \) |
| 47 | \( 1 + 5.41T + 47T^{2} \) |
| 53 | \( 1 - 2.63T + 53T^{2} \) |
| 59 | \( 1 + 1.09T + 59T^{2} \) |
| 61 | \( 1 - 14.7T + 61T^{2} \) |
| 67 | \( 1 + 13.2T + 67T^{2} \) |
| 71 | \( 1 + 2.35T + 71T^{2} \) |
| 73 | \( 1 + 6.45T + 73T^{2} \) |
| 79 | \( 1 - 0.844T + 79T^{2} \) |
| 83 | \( 1 - 13.6T + 83T^{2} \) |
| 89 | \( 1 - 0.307T + 89T^{2} \) |
| 97 | \( 1 + 5.44T + 97T^{2} \) |
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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
−8.182670715756448246634683967910, −7.48879063194255537346088229324, −6.73042881408957125404397568958, −5.97226668033295478477345825750, −5.32189957380200142509118587218, −4.14890886508049147281738041184, −3.41377775538902081064187764220, −3.00322228738002342188200104261, −1.87922848331300222869832455541, −1.07679566745215518805449622944,
1.07679566745215518805449622944, 1.87922848331300222869832455541, 3.00322228738002342188200104261, 3.41377775538902081064187764220, 4.14890886508049147281738041184, 5.32189957380200142509118587218, 5.97226668033295478477345825750, 6.73042881408957125404397568958, 7.48879063194255537346088229324, 8.182670715756448246634683967910
