| L(s) = 1 | − 3-s − 2·4-s − 2·5-s − 5·7-s + 9-s + 2·12-s + 2·15-s + 4·16-s − 2·17-s − 19-s + 4·20-s + 5·21-s − 8·23-s − 25-s − 27-s + 10·28-s − 10·29-s + 3·31-s + 10·35-s − 2·36-s + 11·37-s + 4·41-s + 4·43-s − 2·45-s + 10·47-s − 4·48-s + 18·49-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s − 0.894·5-s − 1.88·7-s + 1/3·9-s + 0.577·12-s + 0.516·15-s + 16-s − 0.485·17-s − 0.229·19-s + 0.894·20-s + 1.09·21-s − 1.66·23-s − 1/5·25-s − 0.192·27-s + 1.88·28-s − 1.85·29-s + 0.538·31-s + 1.69·35-s − 1/3·36-s + 1.80·37-s + 0.624·41-s + 0.609·43-s − 0.298·45-s + 1.45·47-s − 0.577·48-s + 18/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 61347 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61347 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2602688371\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2602688371\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 11 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 11 T + p T^{2} \) |
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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
−14.11625261279266, −13.74728890633058, −12.99465887632880, −12.88511961020837, −12.40209162321130, −11.84782209466672, −11.28987893469824, −10.72620349530478, −10.04042966972623, −9.631731561947955, −9.381819442220335, −8.666475206525457, −8.019041673441933, −7.560784841976601, −6.955642790391520, −6.286535330480391, −5.801466857949414, −5.466873530674654, −4.314012040963655, −4.043715366527698, −3.780602708126799, −2.894844694310180, −2.172333963792192, −0.8635247081121626, −0.2460096164706095,
0.2460096164706095, 0.8635247081121626, 2.172333963792192, 2.894844694310180, 3.780602708126799, 4.043715366527698, 4.314012040963655, 5.466873530674654, 5.801466857949414, 6.286535330480391, 6.955642790391520, 7.560784841976601, 8.019041673441933, 8.666475206525457, 9.381819442220335, 9.631731561947955, 10.04042966972623, 10.72620349530478, 11.28987893469824, 11.84782209466672, 12.40209162321130, 12.88511961020837, 12.99465887632880, 13.74728890633058, 14.11625261279266
