| L(s) = 1 | − 3-s − 2.79·5-s + 0.517·7-s + 9-s − 1.00·11-s − 2.65·13-s + 2.79·15-s − 5.58·17-s − 6.66·19-s − 0.517·21-s − 23-s + 2.82·25-s − 27-s − 29-s − 9.93·31-s + 1.00·33-s − 1.44·35-s − 4.88·37-s + 2.65·39-s + 12.1·41-s + 2.24·43-s − 2.79·45-s − 2.07·47-s − 6.73·49-s + 5.58·51-s − 9.33·53-s + 2.81·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.25·5-s + 0.195·7-s + 0.333·9-s − 0.303·11-s − 0.735·13-s + 0.722·15-s − 1.35·17-s − 1.52·19-s − 0.112·21-s − 0.208·23-s + 0.564·25-s − 0.192·27-s − 0.185·29-s − 1.78·31-s + 0.175·33-s − 0.244·35-s − 0.803·37-s + 0.424·39-s + 1.89·41-s + 0.343·43-s − 0.416·45-s − 0.302·47-s − 0.961·49-s + 0.782·51-s − 1.28·53-s + 0.380·55-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
| \(L(1)\) |
\(\approx\) |
\(0.1816180739\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1816180739\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.79T + 5T^{2} \) |
| 7 | \( 1 - 0.517T + 7T^{2} \) |
| 11 | \( 1 + 1.00T + 11T^{2} \) |
| 13 | \( 1 + 2.65T + 13T^{2} \) |
| 17 | \( 1 + 5.58T + 17T^{2} \) |
| 19 | \( 1 + 6.66T + 19T^{2} \) |
| 31 | \( 1 + 9.93T + 31T^{2} \) |
| 37 | \( 1 + 4.88T + 37T^{2} \) |
| 41 | \( 1 - 12.1T + 41T^{2} \) |
| 43 | \( 1 - 2.24T + 43T^{2} \) |
| 47 | \( 1 + 2.07T + 47T^{2} \) |
| 53 | \( 1 + 9.33T + 53T^{2} \) |
| 59 | \( 1 + 2.47T + 59T^{2} \) |
| 61 | \( 1 - 13.2T + 61T^{2} \) |
| 67 | \( 1 - 10.4T + 67T^{2} \) |
| 71 | \( 1 + 3.17T + 71T^{2} \) |
| 73 | \( 1 + 10.5T + 73T^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 - 4.77T + 83T^{2} \) |
| 89 | \( 1 + 17.2T + 89T^{2} \) |
| 97 | \( 1 - 0.776T + 97T^{2} \) |
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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
−7.76020622362263974691185172835, −7.15717245292372553461849465989, −6.58213714940181249015083719938, −5.73051006551836737228233332563, −4.90541794161851321970463979553, −4.27263457598863744708858701665, −3.81107386081952352638550904257, −2.61734111235578297161401614155, −1.78289983487635010034317997918, −0.20707709735649037527933815802,
0.20707709735649037527933815802, 1.78289983487635010034317997918, 2.61734111235578297161401614155, 3.81107386081952352638550904257, 4.27263457598863744708858701665, 4.90541794161851321970463979553, 5.73051006551836737228233332563, 6.58213714940181249015083719938, 7.15717245292372553461849465989, 7.76020622362263974691185172835
