L(s) = 1 | + 2·3-s + 5-s − 2·7-s + 9-s + 2·13-s + 2·15-s + 6·17-s − 2·19-s − 4·21-s + 25-s − 4·27-s + 6·29-s + 10·31-s − 2·35-s + 37-s + 4·39-s − 6·41-s + 4·43-s + 45-s + 6·47-s − 3·49-s + 12·51-s + 6·53-s − 4·57-s + 6·59-s − 10·61-s − 2·63-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 0.447·5-s − 0.755·7-s + 1/3·9-s + 0.554·13-s + 0.516·15-s + 1.45·17-s − 0.458·19-s − 0.872·21-s + 1/5·25-s − 0.769·27-s + 1.11·29-s + 1.79·31-s − 0.338·35-s + 0.164·37-s + 0.640·39-s − 0.937·41-s + 0.609·43-s + 0.149·45-s + 0.875·47-s − 3/7·49-s + 1.68·51-s + 0.824·53-s − 0.529·57-s + 0.781·59-s − 1.28·61-s − 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.957086323\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.957086323\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 37 | \( 1 - T \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 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
−8.664198216115629432922194533208, −8.191103303293285704368637540717, −7.38307853599763365662944352653, −6.40980423427948262053439128925, −5.87742893251240658386511463008, −4.78046421920466718576600337385, −3.69764756556049564109121180323, −3.06956943298945243015211313058, −2.32883355551645860271045039470, −1.03875136195910076855341646601,
1.03875136195910076855341646601, 2.32883355551645860271045039470, 3.06956943298945243015211313058, 3.69764756556049564109121180323, 4.78046421920466718576600337385, 5.87742893251240658386511463008, 6.40980423427948262053439128925, 7.38307853599763365662944352653, 8.191103303293285704368637540717, 8.664198216115629432922194533208