| L(s) = 1 | − 2.35·2-s + 3.52·4-s + 3.48·7-s − 3.58·8-s − 2.93·11-s − 1.87·13-s − 8.18·14-s + 1.38·16-s − 6.78·17-s − 2.94·19-s + 6.89·22-s + 5.49·23-s + 4.40·26-s + 12.2·28-s − 2.55·29-s + 0.418·31-s + 3.92·32-s + 15.9·34-s + 5.23·37-s + 6.93·38-s − 1.67·41-s + 10.9·43-s − 10.3·44-s − 12.9·46-s + 7.49·47-s + 5.12·49-s − 6.61·52-s + ⋯ |
| L(s) = 1 | − 1.66·2-s + 1.76·4-s + 1.31·7-s − 1.26·8-s − 0.883·11-s − 0.520·13-s − 2.18·14-s + 0.345·16-s − 1.64·17-s − 0.676·19-s + 1.46·22-s + 1.14·23-s + 0.864·26-s + 2.32·28-s − 0.474·29-s + 0.0750·31-s + 0.694·32-s + 2.73·34-s + 0.861·37-s + 1.12·38-s − 0.262·41-s + 1.66·43-s − 1.55·44-s − 1.90·46-s + 1.09·47-s + 0.731·49-s − 0.917·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 2.35T + 2T^{2} \) |
| 7 | \( 1 - 3.48T + 7T^{2} \) |
| 11 | \( 1 + 2.93T + 11T^{2} \) |
| 13 | \( 1 + 1.87T + 13T^{2} \) |
| 17 | \( 1 + 6.78T + 17T^{2} \) |
| 19 | \( 1 + 2.94T + 19T^{2} \) |
| 23 | \( 1 - 5.49T + 23T^{2} \) |
| 29 | \( 1 + 2.55T + 29T^{2} \) |
| 31 | \( 1 - 0.418T + 31T^{2} \) |
| 37 | \( 1 - 5.23T + 37T^{2} \) |
| 41 | \( 1 + 1.67T + 41T^{2} \) |
| 43 | \( 1 - 10.9T + 43T^{2} \) |
| 47 | \( 1 - 7.49T + 47T^{2} \) |
| 53 | \( 1 + 3.70T + 53T^{2} \) |
| 59 | \( 1 - 7.10T + 59T^{2} \) |
| 61 | \( 1 + 6.43T + 61T^{2} \) |
| 67 | \( 1 - 10.0T + 67T^{2} \) |
| 71 | \( 1 + 0.728T + 71T^{2} \) |
| 73 | \( 1 + 3.59T + 73T^{2} \) |
| 79 | \( 1 - 3.07T + 79T^{2} \) |
| 83 | \( 1 + 10.1T + 83T^{2} \) |
| 89 | \( 1 + 0.287T + 89T^{2} \) |
| 97 | \( 1 + 10.4T + 97T^{2} \) |
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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
−7.966339871658517555944272822657, −7.31439059965320077909235270844, −6.79295357119299298598127302032, −5.76368916738547911755014773150, −4.84169355535155121242224728431, −4.25479068982235530507914057487, −2.58486609208314516621314949629, −2.20771924436585126581293763712, −1.15520927151916211763930455195, 0,
1.15520927151916211763930455195, 2.20771924436585126581293763712, 2.58486609208314516621314949629, 4.25479068982235530507914057487, 4.84169355535155121242224728431, 5.76368916738547911755014773150, 6.79295357119299298598127302032, 7.31439059965320077909235270844, 7.966339871658517555944272822657
