| L(s) = 1 | + 0.193·2-s − 1.96·4-s + 7-s − 0.768·8-s − 2·11-s − 1.35·13-s + 0.193·14-s + 3.77·16-s − 3.35·17-s + 5.35·19-s − 0.387·22-s + 4.96·23-s − 0.261·26-s − 1.96·28-s − 7.92·29-s + 4.57·31-s + 2.26·32-s − 0.649·34-s + 0.775·37-s + 1.03·38-s − 3.73·41-s + 12.6·43-s + 3.92·44-s + 0.962·46-s + 9.92·47-s + 49-s + 2.64·52-s + ⋯ |
| L(s) = 1 | + 0.137·2-s − 0.981·4-s + 0.377·7-s − 0.271·8-s − 0.603·11-s − 0.374·13-s + 0.0518·14-s + 0.943·16-s − 0.812·17-s + 1.22·19-s − 0.0826·22-s + 1.03·23-s − 0.0513·26-s − 0.370·28-s − 1.47·29-s + 0.821·31-s + 0.401·32-s − 0.111·34-s + 0.127·37-s + 0.168·38-s − 0.583·41-s + 1.92·43-s + 0.591·44-s + 0.141·46-s + 1.44·47-s + 0.142·49-s + 0.367·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.325890220\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.325890220\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 2 | \( 1 - 0.193T + 2T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 1.35T + 13T^{2} \) |
| 17 | \( 1 + 3.35T + 17T^{2} \) |
| 19 | \( 1 - 5.35T + 19T^{2} \) |
| 23 | \( 1 - 4.96T + 23T^{2} \) |
| 29 | \( 1 + 7.92T + 29T^{2} \) |
| 31 | \( 1 - 4.57T + 31T^{2} \) |
| 37 | \( 1 - 0.775T + 37T^{2} \) |
| 41 | \( 1 + 3.73T + 41T^{2} \) |
| 43 | \( 1 - 12.6T + 43T^{2} \) |
| 47 | \( 1 - 9.92T + 47T^{2} \) |
| 53 | \( 1 - 8.57T + 53T^{2} \) |
| 59 | \( 1 - 8.62T + 59T^{2} \) |
| 61 | \( 1 + 8.70T + 61T^{2} \) |
| 67 | \( 1 + 9.92T + 67T^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 - 9.35T + 73T^{2} \) |
| 79 | \( 1 - 10.7T + 79T^{2} \) |
| 83 | \( 1 - 3.22T + 83T^{2} \) |
| 89 | \( 1 + 1.03T + 89T^{2} \) |
| 97 | \( 1 - 18.4T + 97T^{2} \) |
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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
−9.224472307533169971278752801410, −8.855470603288573011011299565752, −7.76060352040576764076987458162, −7.26506621643004436877254036969, −5.92061764177142006265864474119, −5.19685652811275782803491563851, −4.52238170939592920252372974136, −3.53273732417998985577464912587, −2.41356931023426051398477148174, −0.804343197405415663425383782976,
0.804343197405415663425383782976, 2.41356931023426051398477148174, 3.53273732417998985577464912587, 4.52238170939592920252372974136, 5.19685652811275782803491563851, 5.92061764177142006265864474119, 7.26506621643004436877254036969, 7.76060352040576764076987458162, 8.855470603288573011011299565752, 9.224472307533169971278752801410
