| L(s) = 1 | + 8·11-s + 2·19-s − 16·29-s + 2·31-s − 4·41-s + 13·49-s − 12·59-s + 22·61-s − 12·71-s − 32·79-s − 28·101-s + 22·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 17·169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | + 2.41·11-s + 0.458·19-s − 2.97·29-s + 0.359·31-s − 0.624·41-s + 13/7·49-s − 1.56·59-s + 2.81·61-s − 1.42·71-s − 3.60·79-s − 2.78·101-s + 2.10·109-s + 2.36·121-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 + 1.30·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51840000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51840000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.435825088\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.435825088\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 35 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 53 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 162 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 73 T^{2} + p^{2} T^{4} \) |
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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
−8.365920771510774986461052263567, −7.39888977317734311807190371844, −7.30634452078840110220409203488, −7.26689730276097509124845906834, −6.78235590419781327441395122845, −6.28350636947710007202968528177, −6.04691701856855290574650848911, −5.61037975157108820422932499001, −5.48786382538387658402874709024, −4.88810541639133322263167073716, −4.38282766855184285341524906554, −3.96402449853807106862432092635, −3.95700235499596226824550488418, −3.43905291955878779814683858666, −3.04405358496433305697798548575, −2.43624790737844982539130460300, −1.95241164734278894317745903954, −1.38350717315778280907403745984, −1.27768032425663289028103785433, −0.38776356950331627412301028381,
0.38776356950331627412301028381, 1.27768032425663289028103785433, 1.38350717315778280907403745984, 1.95241164734278894317745903954, 2.43624790737844982539130460300, 3.04405358496433305697798548575, 3.43905291955878779814683858666, 3.95700235499596226824550488418, 3.96402449853807106862432092635, 4.38282766855184285341524906554, 4.88810541639133322263167073716, 5.48786382538387658402874709024, 5.61037975157108820422932499001, 6.04691701856855290574650848911, 6.28350636947710007202968528177, 6.78235590419781327441395122845, 7.26689730276097509124845906834, 7.30634452078840110220409203488, 7.39888977317734311807190371844, 8.365920771510774986461052263567
