L(s) = 1 | − 3·4-s + 5-s + 9-s − 8·11-s + 5·16-s + 8·19-s − 3·20-s + 25-s − 4·29-s − 3·36-s + 20·41-s + 24·44-s + 45-s − 14·49-s − 8·55-s − 8·59-s − 4·61-s − 3·64-s − 16·71-s − 24·76-s + 5·80-s + 81-s − 12·89-s + 8·95-s − 8·99-s − 3·100-s + 12·101-s + ⋯ |
L(s) = 1 | − 3/2·4-s + 0.447·5-s + 1/3·9-s − 2.41·11-s + 5/4·16-s + 1.83·19-s − 0.670·20-s + 1/5·25-s − 0.742·29-s − 1/2·36-s + 3.12·41-s + 3.61·44-s + 0.149·45-s − 2·49-s − 1.07·55-s − 1.04·59-s − 0.512·61-s − 3/8·64-s − 1.89·71-s − 2.75·76-s + 0.559·80-s + 1/9·81-s − 1.27·89-s + 0.820·95-s − 0.804·99-s − 0.299·100-s + 1.19·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1125 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1125 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4386469691\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4386469691\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_1$ | \( 1 - T \) |
good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−14.00692585049802430183598730465, −13.37224542302974911325065415564, −12.89741011334273074809178538613, −12.64617876135106218873747419153, −11.42796116649064334018778191223, −10.67892245123374144028858824682, −9.965958896202437987484594622667, −9.523451675812284265792380668213, −8.843956595114680827452383410006, −7.68003844385765985083943135134, −7.66488013441745380243230842523, −5.86497653842965752491328535183, −5.23920392624592057055772361749, −4.52160444884874669934496061646, −3.01549784140686353642765162463,
3.01549784140686353642765162463, 4.52160444884874669934496061646, 5.23920392624592057055772361749, 5.86497653842965752491328535183, 7.66488013441745380243230842523, 7.68003844385765985083943135134, 8.843956595114680827452383410006, 9.523451675812284265792380668213, 9.965958896202437987484594622667, 10.67892245123374144028858824682, 11.42796116649064334018778191223, 12.64617876135106218873747419153, 12.89741011334273074809178538613, 13.37224542302974911325065415564, 14.00692585049802430183598730465