| L(s) = 1 | − 3-s − 2.85·5-s + 3.46·7-s + 9-s + 1.97·11-s − 6.43·13-s + 2.85·15-s + 0.787·17-s − 0.821·19-s − 3.46·21-s + 4.89·23-s + 3.17·25-s − 27-s − 3.09·29-s + 0.755·31-s − 1.97·33-s − 9.89·35-s + 4.19·37-s + 6.43·39-s − 5.65·41-s + 9.67·43-s − 2.85·45-s + 0.444·47-s + 4.97·49-s − 0.787·51-s − 11.9·53-s − 5.64·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.27·5-s + 1.30·7-s + 0.333·9-s + 0.595·11-s − 1.78·13-s + 0.738·15-s + 0.190·17-s − 0.188·19-s − 0.755·21-s + 1.02·23-s + 0.634·25-s − 0.192·27-s − 0.574·29-s + 0.135·31-s − 0.343·33-s − 1.67·35-s + 0.688·37-s + 1.03·39-s − 0.883·41-s + 1.47·43-s − 0.426·45-s + 0.0648·47-s + 0.711·49-s − 0.110·51-s − 1.64·53-s − 0.760·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 251 | \( 1 - T \) |
| good | 5 | \( 1 + 2.85T + 5T^{2} \) |
| 7 | \( 1 - 3.46T + 7T^{2} \) |
| 11 | \( 1 - 1.97T + 11T^{2} \) |
| 13 | \( 1 + 6.43T + 13T^{2} \) |
| 17 | \( 1 - 0.787T + 17T^{2} \) |
| 19 | \( 1 + 0.821T + 19T^{2} \) |
| 23 | \( 1 - 4.89T + 23T^{2} \) |
| 29 | \( 1 + 3.09T + 29T^{2} \) |
| 31 | \( 1 - 0.755T + 31T^{2} \) |
| 37 | \( 1 - 4.19T + 37T^{2} \) |
| 41 | \( 1 + 5.65T + 41T^{2} \) |
| 43 | \( 1 - 9.67T + 43T^{2} \) |
| 47 | \( 1 - 0.444T + 47T^{2} \) |
| 53 | \( 1 + 11.9T + 53T^{2} \) |
| 59 | \( 1 + 7.73T + 59T^{2} \) |
| 61 | \( 1 + 5.93T + 61T^{2} \) |
| 67 | \( 1 - 2.06T + 67T^{2} \) |
| 71 | \( 1 - 9.58T + 71T^{2} \) |
| 73 | \( 1 + 8.89T + 73T^{2} \) |
| 79 | \( 1 - 15.9T + 79T^{2} \) |
| 83 | \( 1 - 12.2T + 83T^{2} \) |
| 89 | \( 1 + 8.86T + 89T^{2} \) |
| 97 | \( 1 - 1.21T + 97T^{2} \) |
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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
−7.68234626959182933512779545602, −7.23340746440064561196002802514, −6.40671146170624747599303292924, −5.32938694874378726944199544957, −4.73672020110687668125529246982, −4.33129478553192270016907358580, −3.35143645202143703287004171342, −2.24847106129469318531055851047, −1.16123771646102560914385146131, 0,
1.16123771646102560914385146131, 2.24847106129469318531055851047, 3.35143645202143703287004171342, 4.33129478553192270016907358580, 4.73672020110687668125529246982, 5.32938694874378726944199544957, 6.40671146170624747599303292924, 7.23340746440064561196002802514, 7.68234626959182933512779545602
