| L(s) = 1 | + 2.44·2-s + 1.73·3-s + 3.99·4-s + 4.24·6-s + 4.89·8-s + 5·11-s + 6.92·12-s − 1.73·13-s + 3.99·16-s − 1.73·17-s − 2.82·19-s + 12.2·22-s − 2.44·23-s + 8.48·24-s − 4.24·26-s − 5.19·27-s + 5·29-s − 1.41·31-s + 8.66·33-s − 4.24·34-s + 2.44·37-s − 6.92·38-s − 2.99·39-s − 9.89·41-s + 19.9·44-s + ⋯ |
| L(s) = 1 | + 1.73·2-s + 1.00·3-s + 1.99·4-s + 1.73·6-s + 1.73·8-s + 1.50·11-s + 1.99·12-s − 0.480·13-s + 0.999·16-s − 0.420·17-s − 0.648·19-s + 2.61·22-s − 0.510·23-s + 1.73·24-s − 0.832·26-s − 1.00·27-s + 0.928·29-s − 0.254·31-s + 1.50·33-s − 0.727·34-s + 0.402·37-s − 1.12·38-s − 0.480·39-s − 1.54·41-s + 3.01·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.070724349\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.070724349\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 2.44T + 2T^{2} \) |
| 3 | \( 1 - 1.73T + 3T^{2} \) |
| 11 | \( 1 - 5T + 11T^{2} \) |
| 13 | \( 1 + 1.73T + 13T^{2} \) |
| 17 | \( 1 + 1.73T + 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 + 2.44T + 23T^{2} \) |
| 29 | \( 1 - 5T + 29T^{2} \) |
| 31 | \( 1 + 1.41T + 31T^{2} \) |
| 37 | \( 1 - 2.44T + 37T^{2} \) |
| 41 | \( 1 + 9.89T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 8.66T + 47T^{2} \) |
| 53 | \( 1 + 7.34T + 53T^{2} \) |
| 59 | \( 1 - 1.41T + 59T^{2} \) |
| 61 | \( 1 - 2.82T + 61T^{2} \) |
| 67 | \( 1 - 7.34T + 67T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 - 13.8T + 73T^{2} \) |
| 79 | \( 1 + 3T + 79T^{2} \) |
| 83 | \( 1 + 13.8T + 83T^{2} \) |
| 89 | \( 1 - 16.9T + 89T^{2} \) |
| 97 | \( 1 - 1.73T + 97T^{2} \) |
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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
−9.645328637130590430467141263280, −8.857229604579321959014733682879, −8.009126802097124316694234058814, −6.86973026388393531444122524067, −6.37926313289814394298203874451, −5.33129353649935072107092906956, −4.28711361069127727150782959578, −3.72921383621485042806225197959, −2.76974372297008014834243139476, −1.88217596036785265879487630990,
1.88217596036785265879487630990, 2.76974372297008014834243139476, 3.72921383621485042806225197959, 4.28711361069127727150782959578, 5.33129353649935072107092906956, 6.37926313289814394298203874451, 6.86973026388393531444122524067, 8.009126802097124316694234058814, 8.857229604579321959014733682879, 9.645328637130590430467141263280
