| L(s) = 1 | + 5·9-s + 6·11-s + 4·19-s + 18·29-s − 16·31-s + 24·59-s − 16·61-s − 10·79-s + 16·81-s + 24·89-s + 30·99-s + 12·101-s + 38·109-s + 5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 25·169-s + 20·171-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | + 5/3·9-s + 1.80·11-s + 0.917·19-s + 3.34·29-s − 2.87·31-s + 3.12·59-s − 2.04·61-s − 1.12·79-s + 16/9·81-s + 2.54·89-s + 3.01·99-s + 1.19·101-s + 3.63·109-s + 5/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.92·169-s + 1.52·171-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24010000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24010000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.163276104\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.163276104\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 95 T^{2} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.603065637268299122701240434488, −8.067752254235522766690821442662, −7.46597890896270320179846014565, −7.32974718307803183203203355325, −7.14171544229647254250842592286, −6.62033696800988169340629896352, −6.33546966474839103338400339414, −6.10849222096789831357377938109, −5.48197429969527756623327698934, −5.07036069492517075456504060293, −4.58421121154922232442410868444, −4.49710170317929390161925092859, −3.86818284312384878420606600200, −3.58985731513789591328571091730, −3.31651143966793661020085412698, −2.60884281419296112573075834363, −1.98689270840773951893061333010, −1.62176828910906757940563330815, −1.06127112404737090895035348434, −0.74354561630153336456397729461,
0.74354561630153336456397729461, 1.06127112404737090895035348434, 1.62176828910906757940563330815, 1.98689270840773951893061333010, 2.60884281419296112573075834363, 3.31651143966793661020085412698, 3.58985731513789591328571091730, 3.86818284312384878420606600200, 4.49710170317929390161925092859, 4.58421121154922232442410868444, 5.07036069492517075456504060293, 5.48197429969527756623327698934, 6.10849222096789831357377938109, 6.33546966474839103338400339414, 6.62033696800988169340629896352, 7.14171544229647254250842592286, 7.32974718307803183203203355325, 7.46597890896270320179846014565, 8.067752254235522766690821442662, 8.603065637268299122701240434488
