L(s) = 1 | + 0.692·3-s + 2.18·5-s − 7-s − 2.52·9-s − 4.76·11-s + 3.00·13-s + 1.51·15-s + 17-s − 3.29·19-s − 0.692·21-s − 1.05·23-s − 0.225·25-s − 3.82·27-s + 3.24·29-s + 1.23·31-s − 3.29·33-s − 2.18·35-s + 1.17·37-s + 2.07·39-s − 1.21·41-s − 0.149·43-s − 5.50·45-s + 0.529·47-s + 49-s + 0.692·51-s − 6.32·53-s − 10.4·55-s + ⋯ |
L(s) = 1 | + 0.399·3-s + 0.977·5-s − 0.377·7-s − 0.840·9-s − 1.43·11-s + 0.833·13-s + 0.390·15-s + 0.242·17-s − 0.756·19-s − 0.151·21-s − 0.220·23-s − 0.0450·25-s − 0.735·27-s + 0.603·29-s + 0.221·31-s − 0.573·33-s − 0.369·35-s + 0.193·37-s + 0.333·39-s − 0.189·41-s − 0.0228·43-s − 0.821·45-s + 0.0773·47-s + 0.142·49-s + 0.0969·51-s − 0.868·53-s − 1.40·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3808 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 - T \) |
good | 3 | \( 1 - 0.692T + 3T^{2} \) |
| 5 | \( 1 - 2.18T + 5T^{2} \) |
| 11 | \( 1 + 4.76T + 11T^{2} \) |
| 13 | \( 1 - 3.00T + 13T^{2} \) |
| 19 | \( 1 + 3.29T + 19T^{2} \) |
| 23 | \( 1 + 1.05T + 23T^{2} \) |
| 29 | \( 1 - 3.24T + 29T^{2} \) |
| 31 | \( 1 - 1.23T + 31T^{2} \) |
| 37 | \( 1 - 1.17T + 37T^{2} \) |
| 41 | \( 1 + 1.21T + 41T^{2} \) |
| 43 | \( 1 + 0.149T + 43T^{2} \) |
| 47 | \( 1 - 0.529T + 47T^{2} \) |
| 53 | \( 1 + 6.32T + 53T^{2} \) |
| 59 | \( 1 + 2.93T + 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 + 7.61T + 67T^{2} \) |
| 71 | \( 1 + 4.23T + 71T^{2} \) |
| 73 | \( 1 - 0.412T + 73T^{2} \) |
| 79 | \( 1 + 14.5T + 79T^{2} \) |
| 83 | \( 1 - 0.652T + 83T^{2} \) |
| 89 | \( 1 + 13.9T + 89T^{2} \) |
| 97 | \( 1 - 8.93T + 97T^{2} \) |
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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
−8.220199823383204399226348795284, −7.56430243492982655588393743235, −6.41554977878388079724059633879, −5.92745801778416480448149729873, −5.31197308970410456953090090638, −4.29587508293990532033012130487, −3.12230710723398814600782280558, −2.62674609962138086070908280520, −1.63660433969957739827661779609, 0,
1.63660433969957739827661779609, 2.62674609962138086070908280520, 3.12230710723398814600782280558, 4.29587508293990532033012130487, 5.31197308970410456953090090638, 5.92745801778416480448149729873, 6.41554977878388079724059633879, 7.56430243492982655588393743235, 8.220199823383204399226348795284