L(s) = 1 | − 2-s + 3-s + 4-s − 2·5-s − 6-s + 7-s − 8-s + 9-s + 2·10-s + 12-s − 14-s − 2·15-s + 16-s + 2·17-s − 18-s − 4·19-s − 2·20-s + 21-s + 6·23-s − 24-s − 25-s + 27-s + 28-s + 2·30-s − 32-s − 2·34-s − 2·35-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.632·10-s + 0.288·12-s − 0.267·14-s − 0.516·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s − 0.917·19-s − 0.447·20-s + 0.218·21-s + 1.25·23-s − 0.204·24-s − 1/5·25-s + 0.192·27-s + 0.188·28-s + 0.365·30-s − 0.176·32-s − 0.342·34-s − 0.338·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.435973551\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.435973551\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 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
−8.084488634955122203639042440525, −7.33923169686485296770076592158, −6.93265005848127861363526154484, −5.95312479350607585064340468316, −5.03446598373610276044044095510, −4.19549950109638753167292592490, −3.49602858460264210854065523591, −2.66090774135408693189410700411, −1.73825976079137857061092208942, −0.66353415941347160964199031710,
0.66353415941347160964199031710, 1.73825976079137857061092208942, 2.66090774135408693189410700411, 3.49602858460264210854065523591, 4.19549950109638753167292592490, 5.03446598373610276044044095510, 5.95312479350607585064340468316, 6.93265005848127861363526154484, 7.33923169686485296770076592158, 8.084488634955122203639042440525