| L(s) = 1 | + 2.21·3-s + 0.631·5-s + 4.31·7-s + 1.89·9-s − 2.74·11-s − 3.87·13-s + 1.39·15-s + 3.58·17-s + 4.36·19-s + 9.55·21-s − 4.21·23-s − 4.60·25-s − 2.44·27-s + 4.85·29-s + 7.80·31-s − 6.08·33-s + 2.72·35-s − 2.92·37-s − 8.57·39-s + 2.10·41-s + 6.74·43-s + 1.19·45-s + 3.69·47-s + 11.6·49-s + 7.93·51-s − 3.10·53-s − 1.73·55-s + ⋯ |
| L(s) = 1 | + 1.27·3-s + 0.282·5-s + 1.63·7-s + 0.632·9-s − 0.829·11-s − 1.07·13-s + 0.360·15-s + 0.869·17-s + 1.00·19-s + 2.08·21-s − 0.879·23-s − 0.920·25-s − 0.469·27-s + 0.900·29-s + 1.40·31-s − 1.05·33-s + 0.460·35-s − 0.480·37-s − 1.37·39-s + 0.328·41-s + 1.02·43-s + 0.178·45-s + 0.539·47-s + 1.66·49-s + 1.11·51-s − 0.426·53-s − 0.234·55-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
| \(L(1)\) |
\(\approx\) |
\(4.028856386\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.028856386\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.21T + 3T^{2} \) |
| 5 | \( 1 - 0.631T + 5T^{2} \) |
| 7 | \( 1 - 4.31T + 7T^{2} \) |
| 11 | \( 1 + 2.74T + 11T^{2} \) |
| 13 | \( 1 + 3.87T + 13T^{2} \) |
| 17 | \( 1 - 3.58T + 17T^{2} \) |
| 19 | \( 1 - 4.36T + 19T^{2} \) |
| 23 | \( 1 + 4.21T + 23T^{2} \) |
| 29 | \( 1 - 4.85T + 29T^{2} \) |
| 31 | \( 1 - 7.80T + 31T^{2} \) |
| 37 | \( 1 + 2.92T + 37T^{2} \) |
| 41 | \( 1 - 2.10T + 41T^{2} \) |
| 43 | \( 1 - 6.74T + 43T^{2} \) |
| 47 | \( 1 - 3.69T + 47T^{2} \) |
| 53 | \( 1 + 3.10T + 53T^{2} \) |
| 59 | \( 1 - 3.55T + 59T^{2} \) |
| 61 | \( 1 - 14.8T + 61T^{2} \) |
| 67 | \( 1 - 13.5T + 67T^{2} \) |
| 71 | \( 1 - 5.91T + 71T^{2} \) |
| 73 | \( 1 - 10.0T + 73T^{2} \) |
| 79 | \( 1 - 3.57T + 79T^{2} \) |
| 83 | \( 1 + 1.16T + 83T^{2} \) |
| 89 | \( 1 - 1.16T + 89T^{2} \) |
| 97 | \( 1 + 12.5T + 97T^{2} \) |
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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
−8.075032144507678033623867968692, −7.74533463687793066228703627751, −6.98603757646748976733367673492, −5.67426230406660754436069673255, −5.20543966775769307618040763434, −4.42856864438994797303952763095, −3.55509265712701276825336268499, −2.47939780689823275784862543136, −2.22237625704235445977854401747, −1.03332326969489761497856151485,
1.03332326969489761497856151485, 2.22237625704235445977854401747, 2.47939780689823275784862543136, 3.55509265712701276825336268499, 4.42856864438994797303952763095, 5.20543966775769307618040763434, 5.67426230406660754436069673255, 6.98603757646748976733367673492, 7.74533463687793066228703627751, 8.075032144507678033623867968692
