| L(s) = 1 | + 2-s + 3·5-s − 7-s − 8-s + 3·10-s − 3·11-s + 4·13-s − 14-s − 16-s − 12·17-s − 14·19-s − 3·22-s + 3·23-s + 5·25-s + 4·26-s − 5·31-s − 12·34-s − 3·35-s − 14·37-s − 14·38-s − 3·40-s + 9·41-s + 10·43-s + 3·46-s − 6·47-s + 5·50-s + 24·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.34·5-s − 0.377·7-s − 0.353·8-s + 0.948·10-s − 0.904·11-s + 1.10·13-s − 0.267·14-s − 1/4·16-s − 2.91·17-s − 3.21·19-s − 0.639·22-s + 0.625·23-s + 25-s + 0.784·26-s − 0.898·31-s − 2.05·34-s − 0.507·35-s − 2.30·37-s − 2.27·38-s − 0.474·40-s + 1.40·41-s + 1.52·43-s + 0.442·46-s − 0.875·47-s + 0.707·50-s + 3.29·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.124219351\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.124219351\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 9 T + 40 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 10 T + 57 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 10 T + 21 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 8 T - 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−10.21573703022119596023887242263, −9.339331760542930896811336916456, −9.294385969806531741035839376069, −8.682706816228100642796548089709, −8.580036781848298818569007847199, −8.299611877144388942130204100971, −7.23127755651231028803050604544, −6.94665877024398364258631591236, −6.54074369264001809800659256991, −6.16837736391229946838681691161, −5.89162690390824851179670978584, −5.39013433183998221583354016951, −4.64501113420199188984415023813, −4.62988275667730495171662040456, −3.79421652203593157403378953867, −3.61397700878471967169425439184, −2.42810963841338972513826963496, −2.21237225023772979770243576229, −2.05164060815842562480150541531, −0.52708159331006191865992390190,
0.52708159331006191865992390190, 2.05164060815842562480150541531, 2.21237225023772979770243576229, 2.42810963841338972513826963496, 3.61397700878471967169425439184, 3.79421652203593157403378953867, 4.62988275667730495171662040456, 4.64501113420199188984415023813, 5.39013433183998221583354016951, 5.89162690390824851179670978584, 6.16837736391229946838681691161, 6.54074369264001809800659256991, 6.94665877024398364258631591236, 7.23127755651231028803050604544, 8.299611877144388942130204100971, 8.580036781848298818569007847199, 8.682706816228100642796548089709, 9.294385969806531741035839376069, 9.339331760542930896811336916456, 10.21573703022119596023887242263
