L(s) = 1 | + 2·3-s − 3.31·5-s − 3.31·7-s + 9-s − 5·11-s − 6.63·15-s + 5·17-s − 19-s − 6.63·21-s − 6.63·23-s + 6·25-s − 4·27-s − 6.63·29-s − 10·33-s + 11·35-s + 6.63·37-s + 6·41-s + 43-s − 3.31·45-s + 9.94·47-s + 4·49-s + 10·51-s + 13.2·53-s + 16.5·55-s − 2·57-s + 6·59-s − 9.94·61-s + ⋯ |
L(s) = 1 | + 1.15·3-s − 1.48·5-s − 1.25·7-s + 0.333·9-s − 1.50·11-s − 1.71·15-s + 1.21·17-s − 0.229·19-s − 1.44·21-s − 1.38·23-s + 1.20·25-s − 0.769·27-s − 1.23·29-s − 1.74·33-s + 1.85·35-s + 1.09·37-s + 0.937·41-s + 0.152·43-s − 0.494·45-s + 1.45·47-s + 0.571·49-s + 1.40·51-s + 1.82·53-s + 2.23·55-s − 0.264·57-s + 0.781·59-s − 1.27·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9931739466\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9931739466\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 - 2T + 3T^{2} \) |
| 5 | \( 1 + 3.31T + 5T^{2} \) |
| 7 | \( 1 + 3.31T + 7T^{2} \) |
| 11 | \( 1 + 5T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 5T + 17T^{2} \) |
| 23 | \( 1 + 6.63T + 23T^{2} \) |
| 29 | \( 1 + 6.63T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 6.63T + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - T + 43T^{2} \) |
| 47 | \( 1 - 9.94T + 47T^{2} \) |
| 53 | \( 1 - 13.2T + 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 + 9.94T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 - 6.63T + 71T^{2} \) |
| 73 | \( 1 - 9T + 73T^{2} \) |
| 79 | \( 1 + 13.2T + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 - 4T + 89T^{2} \) |
| 97 | \( 1 + 12T + 97T^{2} \) |
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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
−8.131627992596824945468230516855, −7.62367063212017629532932322285, −7.31855882560195778873812215439, −6.05238573055749646551420815496, −5.41126341524580744494476640365, −4.05759765326545220143445358522, −3.74944005304287442355003589359, −2.92463138089217333325510541599, −2.35522274627593311533520259588, −0.48117586178068717928980774477,
0.48117586178068717928980774477, 2.35522274627593311533520259588, 2.92463138089217333325510541599, 3.74944005304287442355003589359, 4.05759765326545220143445358522, 5.41126341524580744494476640365, 6.05238573055749646551420815496, 7.31855882560195778873812215439, 7.62367063212017629532932322285, 8.131627992596824945468230516855