| L(s) = 1 | − 3-s + 1.29·5-s − 7-s + 9-s − 5.01·11-s + 3.71·13-s − 1.29·15-s − 4.41·17-s − 7.71·19-s + 21-s + 2.41·23-s − 3.31·25-s − 27-s − 7.71·29-s − 3.71·31-s + 5.01·33-s − 1.29·35-s + 10.3·37-s − 3.71·39-s + 11.0·41-s + 2.59·43-s + 1.29·45-s + 10.0·47-s + 49-s + 4.41·51-s + 10.3·53-s − 6.51·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.581·5-s − 0.377·7-s + 0.333·9-s − 1.51·11-s + 1.02·13-s − 0.335·15-s − 1.07·17-s − 1.76·19-s + 0.218·21-s + 0.502·23-s − 0.662·25-s − 0.192·27-s − 1.43·29-s − 0.666·31-s + 0.872·33-s − 0.219·35-s + 1.69·37-s − 0.594·39-s + 1.71·41-s + 0.396·43-s + 0.193·45-s + 1.46·47-s + 0.142·49-s + 0.617·51-s + 1.41·53-s − 0.878·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.120843328\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.120843328\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 - 1.29T + 5T^{2} \) |
| 11 | \( 1 + 5.01T + 11T^{2} \) |
| 13 | \( 1 - 3.71T + 13T^{2} \) |
| 17 | \( 1 + 4.41T + 17T^{2} \) |
| 19 | \( 1 + 7.71T + 19T^{2} \) |
| 23 | \( 1 - 2.41T + 23T^{2} \) |
| 29 | \( 1 + 7.71T + 29T^{2} \) |
| 31 | \( 1 + 3.71T + 31T^{2} \) |
| 37 | \( 1 - 10.3T + 37T^{2} \) |
| 41 | \( 1 - 11.0T + 41T^{2} \) |
| 43 | \( 1 - 2.59T + 43T^{2} \) |
| 47 | \( 1 - 10.0T + 47T^{2} \) |
| 53 | \( 1 - 10.3T + 53T^{2} \) |
| 59 | \( 1 - 8.82T + 59T^{2} \) |
| 61 | \( 1 + 9.11T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 + 7.61T + 71T^{2} \) |
| 73 | \( 1 - 12.5T + 73T^{2} \) |
| 79 | \( 1 - 10.5T + 79T^{2} \) |
| 83 | \( 1 - 9.19T + 83T^{2} \) |
| 89 | \( 1 - 4.98T + 89T^{2} \) |
| 97 | \( 1 + 12.0T + 97T^{2} \) |
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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
−8.120414342847235928699442399508, −7.43532671078756032548172752359, −6.60089970207422828455165620538, −5.82965249470624082967503989763, −5.63785801483813711219700072779, −4.44884574885318322983372522256, −3.90980681963584909931578882811, −2.57088737417535729235426554899, −2.04036864979100301169130415702, −0.55983061597845957113495575515,
0.55983061597845957113495575515, 2.04036864979100301169130415702, 2.57088737417535729235426554899, 3.90980681963584909931578882811, 4.44884574885318322983372522256, 5.63785801483813711219700072779, 5.82965249470624082967503989763, 6.60089970207422828455165620538, 7.43532671078756032548172752359, 8.120414342847235928699442399508
