| L(s) = 1 | − 2.35·3-s + 0.425·5-s − 0.578·7-s + 2.54·9-s − 4.03·11-s − 3.33·13-s − 1.00·15-s − 7.84·17-s − 2.14·19-s + 1.36·21-s − 0.686·23-s − 4.81·25-s + 1.07·27-s − 3.07·29-s − 3.74·31-s + 9.50·33-s − 0.246·35-s − 5.44·37-s + 7.84·39-s − 2.94·41-s + 10.8·43-s + 1.08·45-s − 4.16·47-s − 6.66·49-s + 18.4·51-s − 10.0·53-s − 1.71·55-s + ⋯ |
| L(s) = 1 | − 1.35·3-s + 0.190·5-s − 0.218·7-s + 0.848·9-s − 1.21·11-s − 0.924·13-s − 0.258·15-s − 1.90·17-s − 0.493·19-s + 0.297·21-s − 0.143·23-s − 0.963·25-s + 0.206·27-s − 0.571·29-s − 0.672·31-s + 1.65·33-s − 0.0416·35-s − 0.895·37-s + 1.25·39-s − 0.459·41-s + 1.65·43-s + 0.161·45-s − 0.607·47-s − 0.952·49-s + 2.58·51-s − 1.38·53-s − 0.231·55-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
| \(L(1)\) |
\(\approx\) |
\(0.08965856936\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08965856936\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.35T + 3T^{2} \) |
| 5 | \( 1 - 0.425T + 5T^{2} \) |
| 7 | \( 1 + 0.578T + 7T^{2} \) |
| 11 | \( 1 + 4.03T + 11T^{2} \) |
| 13 | \( 1 + 3.33T + 13T^{2} \) |
| 17 | \( 1 + 7.84T + 17T^{2} \) |
| 19 | \( 1 + 2.14T + 19T^{2} \) |
| 23 | \( 1 + 0.686T + 23T^{2} \) |
| 29 | \( 1 + 3.07T + 29T^{2} \) |
| 31 | \( 1 + 3.74T + 31T^{2} \) |
| 37 | \( 1 + 5.44T + 37T^{2} \) |
| 41 | \( 1 + 2.94T + 41T^{2} \) |
| 43 | \( 1 - 10.8T + 43T^{2} \) |
| 47 | \( 1 + 4.16T + 47T^{2} \) |
| 53 | \( 1 + 10.0T + 53T^{2} \) |
| 59 | \( 1 - 8.27T + 59T^{2} \) |
| 61 | \( 1 + 12.2T + 61T^{2} \) |
| 67 | \( 1 + 7.40T + 67T^{2} \) |
| 71 | \( 1 + 1.81T + 71T^{2} \) |
| 73 | \( 1 - 11.6T + 73T^{2} \) |
| 79 | \( 1 - 15.2T + 79T^{2} \) |
| 83 | \( 1 - 7.49T + 83T^{2} \) |
| 89 | \( 1 + 0.487T + 89T^{2} \) |
| 97 | \( 1 - 5.00T + 97T^{2} \) |
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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
−7.88543949317061464447139693653, −7.28064231878014984068601658659, −6.43458095047959749938619461330, −6.05162024440969683867524908511, −5.05576482092414761627461096574, −4.86258925403505738715099629143, −3.82343033960068259491558542426, −2.58393475864575632526480262606, −1.87981543446501136172973855900, −0.15724278915646547663713574544,
0.15724278915646547663713574544, 1.87981543446501136172973855900, 2.58393475864575632526480262606, 3.82343033960068259491558542426, 4.86258925403505738715099629143, 5.05576482092414761627461096574, 6.05162024440969683867524908511, 6.43458095047959749938619461330, 7.28064231878014984068601658659, 7.88543949317061464447139693653
