| L(s) = 1 | − 3·3-s + 5·4-s − 6·5-s + 3·11-s − 15·12-s + 4·13-s + 18·15-s + 9·16-s + 27·17-s − 11·19-s − 30·20-s + 48·23-s − 25-s + 27·27-s + 78·29-s + 64·31-s − 9·33-s + 34·37-s − 12·39-s − 21·41-s + 61·43-s + 15·44-s − 27·48-s − 81·51-s + 20·52-s − 18·55-s + 33·57-s + ⋯ |
| L(s) = 1 | − 3-s + 5/4·4-s − 6/5·5-s + 3/11·11-s − 5/4·12-s + 4/13·13-s + 6/5·15-s + 9/16·16-s + 1.58·17-s − 0.578·19-s − 3/2·20-s + 2.08·23-s − 0.0399·25-s + 27-s + 2.68·29-s + 2.06·31-s − 0.272·33-s + 0.918·37-s − 0.307·39-s − 0.512·41-s + 1.41·43-s + 0.340·44-s − 0.562·48-s − 1.58·51-s + 5/13·52-s − 0.327·55-s + 0.578·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.131738187\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.131738187\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 5 T^{2} + p^{4} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 6 T + 37 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 3 T + 124 T^{2} - 3 p^{2} T^{3} + p^{4} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 4 T - 153 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 27 T + 532 T^{2} - 27 p^{2} T^{3} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 26 T + p^{2} T^{2} )( 1 + 37 T + p^{2} T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 48 T + 1297 T^{2} - 48 p^{2} T^{3} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 78 T + 2869 T^{2} - 78 p^{2} T^{3} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 32 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 34 T - 213 T^{2} - 34 p^{2} T^{3} + p^{4} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 21 T + 1828 T^{2} + 21 p^{2} T^{3} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 83 T + p^{2} T^{2} )( 1 + 22 T + p^{2} T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 2066 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 4439 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 56 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 31 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 9110 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 65 T - 1104 T^{2} + 65 p^{2} T^{3} + p^{4} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 38 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 84 T + 9241 T^{2} + 84 p^{2} T^{3} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 216 T + 23473 T^{2} + 216 p^{2} T^{3} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 115 T + 3816 T^{2} - 115 p^{2} T^{3} + p^{4} T^{4} \) |
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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
−11.30704143673092850690197596530, −10.79099681125557011835123606332, −10.34137919385693955204457431290, −10.16994052149051360475148903293, −9.383088613660855499231476493111, −8.714025859843364598546306762638, −8.116718544770657019058916020839, −8.115565213032866939296626031528, −7.22953465595280053776904021674, −6.93678753518574773483949803314, −6.52590256006308433673962941432, −5.99835311765873498431896863198, −5.61831408657135468014299477922, −4.65206702712917515910319057020, −4.60091723265448127653998836105, −3.64508853238292063849600193329, −2.78506120359528270258312601508, −2.77498822113853578336385968400, −1.16713103074793265534927428255, −0.791861851556797233383498870812,
0.791861851556797233383498870812, 1.16713103074793265534927428255, 2.77498822113853578336385968400, 2.78506120359528270258312601508, 3.64508853238292063849600193329, 4.60091723265448127653998836105, 4.65206702712917515910319057020, 5.61831408657135468014299477922, 5.99835311765873498431896863198, 6.52590256006308433673962941432, 6.93678753518574773483949803314, 7.22953465595280053776904021674, 8.115565213032866939296626031528, 8.116718544770657019058916020839, 8.714025859843364598546306762638, 9.383088613660855499231476493111, 10.16994052149051360475148903293, 10.34137919385693955204457431290, 10.79099681125557011835123606332, 11.30704143673092850690197596530
