| L(s) = 1 | + 7-s + 4·9-s + 6·11-s − 2·23-s + 2·25-s + 8·29-s − 4·37-s + 14·43-s + 49-s − 10·53-s + 4·63-s + 16·71-s + 6·77-s + 4·79-s + 7·81-s + 24·99-s − 4·107-s + 8·109-s + 16·113-s + 14·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | + 0.377·7-s + 4/3·9-s + 1.80·11-s − 0.417·23-s + 2/5·25-s + 1.48·29-s − 0.657·37-s + 2.13·43-s + 1/7·49-s − 1.37·53-s + 0.503·63-s + 1.89·71-s + 0.683·77-s + 0.450·79-s + 7/9·81-s + 2.41·99-s − 0.386·107-s + 0.766·109-s + 1.50·113-s + 1.27·121-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1404928 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1404928 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.344042057\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.344042057\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( 1 - T \) |
| good | 3 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 28 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 44 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 68 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
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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
−7.88271999054956306888307232270, −7.45943982376063667814135562227, −7.04761845591218427151143510475, −6.69144689829838502634440515341, −6.17101789862618612978276346448, −6.04912882524166439112755959402, −5.05705532539122247573231962184, −4.83820438444528748757849533316, −4.29264292764809553400432414941, −3.88319088519490916872096187583, −3.53792161962180118554097474639, −2.69225121398362659532898766006, −2.03813926427917620595361375155, −1.35280166475576510502760033632, −0.930323697563573287203759877328,
0.930323697563573287203759877328, 1.35280166475576510502760033632, 2.03813926427917620595361375155, 2.69225121398362659532898766006, 3.53792161962180118554097474639, 3.88319088519490916872096187583, 4.29264292764809553400432414941, 4.83820438444528748757849533316, 5.05705532539122247573231962184, 6.04912882524166439112755959402, 6.17101789862618612978276346448, 6.69144689829838502634440515341, 7.04761845591218427151143510475, 7.45943982376063667814135562227, 7.88271999054956306888307232270
