L(s) = 1 | + 3-s + 2·5-s + 9-s + 4·11-s + 2·13-s + 2·15-s − 2·17-s + 4·19-s − 8·23-s − 25-s + 27-s + 6·29-s − 8·31-s + 4·33-s + 6·37-s + 2·39-s + 6·41-s + 4·43-s + 2·45-s − 2·51-s − 2·53-s + 8·55-s + 4·57-s − 4·59-s + 2·61-s + 4·65-s − 4·67-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.894·5-s + 1/3·9-s + 1.20·11-s + 0.554·13-s + 0.516·15-s − 0.485·17-s + 0.917·19-s − 1.66·23-s − 1/5·25-s + 0.192·27-s + 1.11·29-s − 1.43·31-s + 0.696·33-s + 0.986·37-s + 0.320·39-s + 0.937·41-s + 0.609·43-s + 0.298·45-s − 0.280·51-s − 0.274·53-s + 1.07·55-s + 0.529·57-s − 0.520·59-s + 0.256·61-s + 0.496·65-s − 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.548614977\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.548614977\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−9.541792988687802337799065273854, −9.172969029955001041869475015652, −8.228097915824748265448934137029, −7.33253412720106144652855023371, −6.30552876175851347232907272447, −5.77649722095232052155724795965, −4.41794336687255381402125784080, −3.60246495583988551500687864369, −2.34965244557392920909834502109, −1.35556028534757571668534423075,
1.35556028534757571668534423075, 2.34965244557392920909834502109, 3.60246495583988551500687864369, 4.41794336687255381402125784080, 5.77649722095232052155724795965, 6.30552876175851347232907272447, 7.33253412720106144652855023371, 8.228097915824748265448934137029, 9.172969029955001041869475015652, 9.541792988687802337799065273854