| L(s) = 1 | + 3-s + 0.864·5-s + 2.62·7-s + 9-s − 0.626·11-s − 2.62·13-s + 0.864·15-s + 7.28·17-s + 1.10·19-s + 2.62·21-s + 2.23·23-s − 4.25·25-s + 27-s − 0.864·29-s + 3.52·31-s − 0.626·33-s + 2.27·35-s − 37-s − 2.62·39-s + 2·41-s − 8.77·43-s + 0.864·45-s + 5.79·47-s − 0.103·49-s + 7.28·51-s − 9.67·53-s − 0.541·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.386·5-s + 0.992·7-s + 0.333·9-s − 0.188·11-s − 0.728·13-s + 0.223·15-s + 1.76·17-s + 0.253·19-s + 0.573·21-s + 0.466·23-s − 0.850·25-s + 0.192·27-s − 0.160·29-s + 0.632·31-s − 0.109·33-s + 0.383·35-s − 0.164·37-s − 0.420·39-s + 0.312·41-s − 1.33·43-s + 0.128·45-s + 0.845·47-s − 0.0147·49-s + 1.02·51-s − 1.32·53-s − 0.0730·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.295981462\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.295981462\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 37 | \( 1 + T \) |
| good | 5 | \( 1 - 0.864T + 5T^{2} \) |
| 7 | \( 1 - 2.62T + 7T^{2} \) |
| 11 | \( 1 + 0.626T + 11T^{2} \) |
| 13 | \( 1 + 2.62T + 13T^{2} \) |
| 17 | \( 1 - 7.28T + 17T^{2} \) |
| 19 | \( 1 - 1.10T + 19T^{2} \) |
| 23 | \( 1 - 2.23T + 23T^{2} \) |
| 29 | \( 1 + 0.864T + 29T^{2} \) |
| 31 | \( 1 - 3.52T + 31T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 8.77T + 43T^{2} \) |
| 47 | \( 1 - 5.79T + 47T^{2} \) |
| 53 | \( 1 + 9.67T + 53T^{2} \) |
| 59 | \( 1 - 12.6T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 - 10.7T + 67T^{2} \) |
| 71 | \( 1 - 3.45T + 71T^{2} \) |
| 73 | \( 1 - 9.40T + 73T^{2} \) |
| 79 | \( 1 - 1.79T + 79T^{2} \) |
| 83 | \( 1 - 2.83T + 83T^{2} \) |
| 89 | \( 1 + 7.22T + 89T^{2} \) |
| 97 | \( 1 + 7.31T + 97T^{2} \) |
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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
−9.918043158220811011581609099763, −9.469719381088934251166208540087, −8.193487400926363073930770362936, −7.85971974215387134769984976704, −6.86105190671014177631762574141, −5.55941504287958249026119584830, −4.91861909142671580520631121734, −3.67420189524042721863334946962, −2.53717103312983894031604439482, −1.37778908604533572062220680173,
1.37778908604533572062220680173, 2.53717103312983894031604439482, 3.67420189524042721863334946962, 4.91861909142671580520631121734, 5.55941504287958249026119584830, 6.86105190671014177631762574141, 7.85971974215387134769984976704, 8.193487400926363073930770362936, 9.469719381088934251166208540087, 9.918043158220811011581609099763
