| L(s) = 1 | + 3-s − 2·4-s + 7-s + 9-s − 2·12-s − 13-s + 12·19-s + 21-s − 2·25-s + 27-s − 2·28-s − 6·31-s − 2·36-s − 37-s − 39-s + 4·43-s + 49-s + 2·52-s + 12·57-s + 5·61-s + 63-s + 8·64-s + 17·67-s + 8·73-s − 2·75-s − 24·76-s + 20·79-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s + 0.377·7-s + 1/3·9-s − 0.577·12-s − 0.277·13-s + 2.75·19-s + 0.218·21-s − 2/5·25-s + 0.192·27-s − 0.377·28-s − 1.07·31-s − 1/3·36-s − 0.164·37-s − 0.160·39-s + 0.609·43-s + 1/7·49-s + 0.277·52-s + 1.58·57-s + 0.640·61-s + 0.125·63-s + 64-s + 2.07·67-s + 0.936·73-s − 0.230·75-s − 2.75·76-s + 2.25·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 120393 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120393 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.584104418\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.584104418\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 7 | $C_1$ | \( 1 - T \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 79 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 100 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 124 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−9.395486606312315156765858977607, −9.139305917373504230961163512576, −8.476262644678221203518185183962, −7.902711381126855865472472686220, −7.72283714297968298578527439604, −7.03501453412470613960642073402, −6.62761803277701905122981498512, −5.47361219511936212758832352865, −5.40047333071869439240791326828, −4.83983205558822558394388037842, −4.01821765642012429102241441390, −3.64752210011396241713028297417, −2.90270195030581545310145274007, −2.02658998755924228929837049572, −0.922960980672013519887507722604,
0.922960980672013519887507722604, 2.02658998755924228929837049572, 2.90270195030581545310145274007, 3.64752210011396241713028297417, 4.01821765642012429102241441390, 4.83983205558822558394388037842, 5.40047333071869439240791326828, 5.47361219511936212758832352865, 6.62761803277701905122981498512, 7.03501453412470613960642073402, 7.72283714297968298578527439604, 7.902711381126855865472472686220, 8.476262644678221203518185183962, 9.139305917373504230961163512576, 9.395486606312315156765858977607
