| L(s) = 1 | − 4·9-s − 8·11-s − 2·23-s − 2·25-s − 12·29-s + 4·37-s + 12·53-s + 8·67-s − 16·71-s − 8·79-s + 7·81-s + 32·99-s + 20·109-s − 12·113-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 8·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | − 4/3·9-s − 2.41·11-s − 0.417·23-s − 2/5·25-s − 2.22·29-s + 0.657·37-s + 1.64·53-s + 0.977·67-s − 1.89·71-s − 0.900·79-s + 7/9·81-s + 3.21·99-s + 1.91·109-s − 1.12·113-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 − 0.615·169-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81288256 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81288256 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 + T )^{2} \) |
| 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 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 60 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 80 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 68 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 116 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 144 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 134 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 146 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
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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
−7.58988437783145025945604719492, −7.41486769371745218533805302584, −6.97132948876807652965424348209, −6.52150904412741973907188179426, −6.00789102842929801154970672264, −5.66952679577236827532227405572, −5.52097618366204639020407813846, −5.44720350580673104693291726142, −4.74537158926309209151644486259, −4.58772088293201353554949632620, −3.89382659123186735514414109793, −3.70940087382509021647162499906, −3.08066857894006727600329719202, −2.87973608099609726247484673523, −2.35212580895531066614362641244, −2.21061349851071301407462870015, −1.62913620649332960643332714830, −0.791040424129589394551948793566, 0, 0,
0.791040424129589394551948793566, 1.62913620649332960643332714830, 2.21061349851071301407462870015, 2.35212580895531066614362641244, 2.87973608099609726247484673523, 3.08066857894006727600329719202, 3.70940087382509021647162499906, 3.89382659123186735514414109793, 4.58772088293201353554949632620, 4.74537158926309209151644486259, 5.44720350580673104693291726142, 5.52097618366204639020407813846, 5.66952679577236827532227405572, 6.00789102842929801154970672264, 6.52150904412741973907188179426, 6.97132948876807652965424348209, 7.41486769371745218533805302584, 7.58988437783145025945604719492
