| L(s) = 1 | + 2·3-s + 3·9-s − 8·19-s − 10·25-s + 4·27-s − 12·29-s − 16·31-s + 4·37-s − 16·47-s − 4·53-s − 16·57-s − 8·59-s − 20·75-s + 5·81-s + 24·83-s − 24·87-s − 32·93-s − 16·103-s − 12·109-s + 8·111-s + 4·113-s + 10·121-s + 127-s + 131-s + 137-s + 139-s − 32·141-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 9-s − 1.83·19-s − 2·25-s + 0.769·27-s − 2.22·29-s − 2.87·31-s + 0.657·37-s − 2.33·47-s − 0.549·53-s − 2.11·57-s − 1.04·59-s − 2.30·75-s + 5/9·81-s + 2.63·83-s − 2.57·87-s − 3.31·93-s − 1.57·103-s − 1.14·109-s + 0.759·111-s + 0.376·113-s + 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2.69·141-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22127616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22127616 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 146 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
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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
−8.018902320235226237984161948779, −7.86803879564576230678808804028, −7.52602732555795333172625892010, −7.19253141449323505403691123047, −6.67067560218068380721315713147, −6.41131904302655218425961032573, −5.84492654958310546434524143885, −5.70868807662515603735047494202, −5.11364714260986217407319531433, −4.71523697857490564674101301835, −4.21650273077960255555017086124, −3.82328279697478536046455080738, −3.52791634479957408774953434472, −3.37373011527687384222043684565, −2.36525829200758333274337660851, −2.30141250783538983809366869523, −1.64316166120515824498230239649, −1.58798917508129634673207512628, 0, 0,
1.58798917508129634673207512628, 1.64316166120515824498230239649, 2.30141250783538983809366869523, 2.36525829200758333274337660851, 3.37373011527687384222043684565, 3.52791634479957408774953434472, 3.82328279697478536046455080738, 4.21650273077960255555017086124, 4.71523697857490564674101301835, 5.11364714260986217407319531433, 5.70868807662515603735047494202, 5.84492654958310546434524143885, 6.41131904302655218425961032573, 6.67067560218068380721315713147, 7.19253141449323505403691123047, 7.52602732555795333172625892010, 7.86803879564576230678808804028, 8.018902320235226237984161948779
