| L(s) = 1 | − 3-s + 0.0891·5-s + 2.52·7-s + 9-s − 1.55·11-s + 6.31·13-s − 0.0891·15-s − 6.02·17-s + 4.11·19-s − 2.52·21-s + 23-s − 4.99·25-s − 27-s − 29-s − 6.28·31-s + 1.55·33-s + 0.224·35-s − 3.14·37-s − 6.31·39-s − 3.88·41-s + 1.91·43-s + 0.0891·45-s − 12.5·47-s − 0.645·49-s + 6.02·51-s − 3.50·53-s − 0.138·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.0398·5-s + 0.952·7-s + 0.333·9-s − 0.467·11-s + 1.75·13-s − 0.0230·15-s − 1.46·17-s + 0.945·19-s − 0.550·21-s + 0.208·23-s − 0.998·25-s − 0.192·27-s − 0.185·29-s − 1.12·31-s + 0.270·33-s + 0.0379·35-s − 0.516·37-s − 1.01·39-s − 0.606·41-s + 0.292·43-s + 0.0132·45-s − 1.82·47-s − 0.0921·49-s + 0.843·51-s − 0.481·53-s − 0.0186·55-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 5 | \( 1 - 0.0891T + 5T^{2} \) |
| 7 | \( 1 - 2.52T + 7T^{2} \) |
| 11 | \( 1 + 1.55T + 11T^{2} \) |
| 13 | \( 1 - 6.31T + 13T^{2} \) |
| 17 | \( 1 + 6.02T + 17T^{2} \) |
| 19 | \( 1 - 4.11T + 19T^{2} \) |
| 31 | \( 1 + 6.28T + 31T^{2} \) |
| 37 | \( 1 + 3.14T + 37T^{2} \) |
| 41 | \( 1 + 3.88T + 41T^{2} \) |
| 43 | \( 1 - 1.91T + 43T^{2} \) |
| 47 | \( 1 + 12.5T + 47T^{2} \) |
| 53 | \( 1 + 3.50T + 53T^{2} \) |
| 59 | \( 1 + 9.16T + 59T^{2} \) |
| 61 | \( 1 - 10.3T + 61T^{2} \) |
| 67 | \( 1 + 1.85T + 67T^{2} \) |
| 71 | \( 1 - 2.06T + 71T^{2} \) |
| 73 | \( 1 + 0.852T + 73T^{2} \) |
| 79 | \( 1 + 14.6T + 79T^{2} \) |
| 83 | \( 1 - 1.87T + 83T^{2} \) |
| 89 | \( 1 + 11.8T + 89T^{2} \) |
| 97 | \( 1 + 5.06T + 97T^{2} \) |
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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
−7.48776369901954375924835746377, −6.74207295403015515157052303689, −6.04568748627271719976862442418, −5.40341377788925110690124710066, −4.76757011670533176639862710531, −3.98777231361848310031508296191, −3.22509120441562922238909594297, −1.94039604880786427856304263683, −1.36880932160754387390361150958, 0,
1.36880932160754387390361150958, 1.94039604880786427856304263683, 3.22509120441562922238909594297, 3.98777231361848310031508296191, 4.76757011670533176639862710531, 5.40341377788925110690124710066, 6.04568748627271719976862442418, 6.74207295403015515157052303689, 7.48776369901954375924835746377
