| L(s) = 1 | − 156·3-s − 952·7-s + 4.65e3·9-s + 5.61e4·11-s − 1.78e5·13-s + 2.47e5·17-s − 3.15e5·19-s + 1.48e5·21-s + 2.04e5·23-s + 2.34e6·27-s − 3.84e6·29-s + 1.30e6·31-s − 8.75e6·33-s − 4.30e6·37-s + 2.77e7·39-s + 1.51e6·41-s + 3.36e7·43-s − 1.05e7·47-s − 3.94e7·49-s − 3.86e7·51-s − 1.66e7·53-s + 4.91e7·57-s − 1.12e8·59-s − 3.31e7·61-s − 4.42e6·63-s − 1.21e8·67-s − 3.19e7·69-s + ⋯ |
| L(s) = 1 | − 1.11·3-s − 0.149·7-s + 0.236·9-s + 1.15·11-s − 1.72·13-s + 0.719·17-s − 0.555·19-s + 0.166·21-s + 0.152·23-s + 0.849·27-s − 1.00·29-s + 0.254·31-s − 1.28·33-s − 0.377·37-s + 1.92·39-s + 0.0835·41-s + 1.50·43-s − 0.316·47-s − 0.977·49-s − 0.799·51-s − 0.289·53-s + 0.617·57-s − 1.20·59-s − 0.306·61-s − 0.0354·63-s − 0.735·67-s − 0.169·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.7585577238\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7585577238\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 52 p T + p^{9} T^{2} \) |
| 7 | \( 1 + 136 p T + p^{9} T^{2} \) |
| 11 | \( 1 - 56148 T + p^{9} T^{2} \) |
| 13 | \( 1 + 178094 T + p^{9} T^{2} \) |
| 17 | \( 1 - 247662 T + p^{9} T^{2} \) |
| 19 | \( 1 + 315380 T + p^{9} T^{2} \) |
| 23 | \( 1 - 204504 T + p^{9} T^{2} \) |
| 29 | \( 1 + 3840450 T + p^{9} T^{2} \) |
| 31 | \( 1 - 1309408 T + p^{9} T^{2} \) |
| 37 | \( 1 + 4307078 T + p^{9} T^{2} \) |
| 41 | \( 1 - 1512042 T + p^{9} T^{2} \) |
| 43 | \( 1 - 33670604 T + p^{9} T^{2} \) |
| 47 | \( 1 + 10581072 T + p^{9} T^{2} \) |
| 53 | \( 1 + 16616214 T + p^{9} T^{2} \) |
| 59 | \( 1 + 112235100 T + p^{9} T^{2} \) |
| 61 | \( 1 + 33197218 T + p^{9} T^{2} \) |
| 67 | \( 1 + 121372252 T + p^{9} T^{2} \) |
| 71 | \( 1 - 387172728 T + p^{9} T^{2} \) |
| 73 | \( 1 + 255240074 T + p^{9} T^{2} \) |
| 79 | \( 1 + 492101840 T + p^{9} T^{2} \) |
| 83 | \( 1 + 457420236 T + p^{9} T^{2} \) |
| 89 | \( 1 + 31809510 T + p^{9} T^{2} \) |
| 97 | \( 1 - 673532062 T + p^{9} T^{2} \) |
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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
−9.819331812197313602179147978735, −9.062226338151572034655191755909, −7.71910665758332246618867177527, −6.83106795503963308351841122802, −5.99229410594189541620717959497, −5.11032443697653954363255792702, −4.20017627093093187352436045699, −2.87316618721685206628990991634, −1.55756107097812144432064839510, −0.39082757226057019190633769882,
0.39082757226057019190633769882, 1.55756107097812144432064839510, 2.87316618721685206628990991634, 4.20017627093093187352436045699, 5.11032443697653954363255792702, 5.99229410594189541620717959497, 6.83106795503963308351841122802, 7.71910665758332246618867177527, 9.062226338151572034655191755909, 9.819331812197313602179147978735
