| L(s) = 1 | − 3-s + 4.44·5-s − 0.0380·7-s + 9-s + 4.85·11-s + 1.07·13-s − 4.44·15-s + 2.50·17-s − 0.902·19-s + 0.0380·21-s − 23-s + 14.7·25-s − 27-s − 29-s − 3.84·31-s − 4.85·33-s − 0.169·35-s + 8.10·37-s − 1.07·39-s + 1.14·41-s − 7.38·43-s + 4.44·45-s + 5.54·47-s − 6.99·49-s − 2.50·51-s + 6.54·53-s + 21.5·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.98·5-s − 0.0143·7-s + 0.333·9-s + 1.46·11-s + 0.299·13-s − 1.14·15-s + 0.608·17-s − 0.207·19-s + 0.00831·21-s − 0.208·23-s + 2.94·25-s − 0.192·27-s − 0.185·29-s − 0.690·31-s − 0.845·33-s − 0.0286·35-s + 1.33·37-s − 0.172·39-s + 0.178·41-s − 1.12·43-s + 0.662·45-s + 0.808·47-s − 0.999·49-s − 0.351·51-s + 0.899·53-s + 2.90·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.176192293\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.176192293\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 5 | \( 1 - 4.44T + 5T^{2} \) |
| 7 | \( 1 + 0.0380T + 7T^{2} \) |
| 11 | \( 1 - 4.85T + 11T^{2} \) |
| 13 | \( 1 - 1.07T + 13T^{2} \) |
| 17 | \( 1 - 2.50T + 17T^{2} \) |
| 19 | \( 1 + 0.902T + 19T^{2} \) |
| 31 | \( 1 + 3.84T + 31T^{2} \) |
| 37 | \( 1 - 8.10T + 37T^{2} \) |
| 41 | \( 1 - 1.14T + 41T^{2} \) |
| 43 | \( 1 + 7.38T + 43T^{2} \) |
| 47 | \( 1 - 5.54T + 47T^{2} \) |
| 53 | \( 1 - 6.54T + 53T^{2} \) |
| 59 | \( 1 + 1.34T + 59T^{2} \) |
| 61 | \( 1 - 6.90T + 61T^{2} \) |
| 67 | \( 1 - 9.06T + 67T^{2} \) |
| 71 | \( 1 + 1.52T + 71T^{2} \) |
| 73 | \( 1 + 1.88T + 73T^{2} \) |
| 79 | \( 1 + 7.21T + 79T^{2} \) |
| 83 | \( 1 - 0.623T + 83T^{2} \) |
| 89 | \( 1 - 0.312T + 89T^{2} \) |
| 97 | \( 1 - 15.0T + 97T^{2} \) |
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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
−7.75151509747774088746314211590, −6.67258264525389444595615577994, −6.52106332731299529153711469710, −5.73427769249162544573570918537, −5.33720230948685424795752764761, −4.40046197285811669682560040344, −3.53663704922489622092679122657, −2.45941729246748754348597458231, −1.63218573171740813127080778984, −1.01181411307990213117552531250,
1.01181411307990213117552531250, 1.63218573171740813127080778984, 2.45941729246748754348597458231, 3.53663704922489622092679122657, 4.40046197285811669682560040344, 5.33720230948685424795752764761, 5.73427769249162544573570918537, 6.52106332731299529153711469710, 6.67258264525389444595615577994, 7.75151509747774088746314211590
