| L(s) = 1 | − 3.31·3-s − 2.21·5-s + 1.66·7-s + 7.99·9-s + 4.05·11-s + 7.06·13-s + 7.34·15-s − 0.849·17-s − 0.238·19-s − 5.53·21-s − 7.52·23-s − 0.0998·25-s − 16.5·27-s + 5.42·29-s − 4.08·31-s − 13.4·33-s − 3.69·35-s + 3.45·37-s − 23.4·39-s + 2.76·41-s − 10.3·43-s − 17.6·45-s + 2.66·47-s − 4.21·49-s + 2.81·51-s − 2.93·53-s − 8.97·55-s + ⋯ |
| L(s) = 1 | − 1.91·3-s − 0.989·5-s + 0.630·7-s + 2.66·9-s + 1.22·11-s + 1.95·13-s + 1.89·15-s − 0.206·17-s − 0.0547·19-s − 1.20·21-s − 1.56·23-s − 0.0199·25-s − 3.18·27-s + 1.00·29-s − 0.733·31-s − 2.34·33-s − 0.624·35-s + 0.567·37-s − 3.75·39-s + 0.432·41-s − 1.58·43-s − 2.63·45-s + 0.388·47-s − 0.602·49-s + 0.394·51-s − 0.402·53-s − 1.21·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 503 | \( 1 - T \) |
| good | 3 | \( 1 + 3.31T + 3T^{2} \) |
| 5 | \( 1 + 2.21T + 5T^{2} \) |
| 7 | \( 1 - 1.66T + 7T^{2} \) |
| 11 | \( 1 - 4.05T + 11T^{2} \) |
| 13 | \( 1 - 7.06T + 13T^{2} \) |
| 17 | \( 1 + 0.849T + 17T^{2} \) |
| 19 | \( 1 + 0.238T + 19T^{2} \) |
| 23 | \( 1 + 7.52T + 23T^{2} \) |
| 29 | \( 1 - 5.42T + 29T^{2} \) |
| 31 | \( 1 + 4.08T + 31T^{2} \) |
| 37 | \( 1 - 3.45T + 37T^{2} \) |
| 41 | \( 1 - 2.76T + 41T^{2} \) |
| 43 | \( 1 + 10.3T + 43T^{2} \) |
| 47 | \( 1 - 2.66T + 47T^{2} \) |
| 53 | \( 1 + 2.93T + 53T^{2} \) |
| 59 | \( 1 - 11.4T + 59T^{2} \) |
| 61 | \( 1 + 7.26T + 61T^{2} \) |
| 67 | \( 1 + 2.16T + 67T^{2} \) |
| 71 | \( 1 + 9.94T + 71T^{2} \) |
| 73 | \( 1 - 1.67T + 73T^{2} \) |
| 79 | \( 1 + 10.0T + 79T^{2} \) |
| 83 | \( 1 - 1.02T + 83T^{2} \) |
| 89 | \( 1 + 1.18T + 89T^{2} \) |
| 97 | \( 1 + 14.4T + 97T^{2} \) |
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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
−7.32292688352007564264336817392, −6.54258420033892551716621349118, −6.16202480952256818647691270339, −5.54629426201956344125735688436, −4.54790580327902829131785732075, −4.11800151184937862315649534210, −3.59147848845310582494332128293, −1.67442566275759612423259703199, −1.11886263848923086280221203128, 0,
1.11886263848923086280221203128, 1.67442566275759612423259703199, 3.59147848845310582494332128293, 4.11800151184937862315649534210, 4.54790580327902829131785732075, 5.54629426201956344125735688436, 6.16202480952256818647691270339, 6.54258420033892551716621349118, 7.32292688352007564264336817392
