L(s) = 1 | − 3-s − 3·4-s + 4·5-s − 7-s + 9-s + 3·12-s − 4·15-s + 5·16-s + 12·17-s − 12·20-s + 21-s + 2·25-s − 27-s + 3·28-s − 4·35-s − 3·36-s + 12·37-s − 4·41-s − 8·43-s + 4·45-s − 5·48-s + 49-s − 12·51-s − 24·59-s + 12·60-s − 63-s − 3·64-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 3/2·4-s + 1.78·5-s − 0.377·7-s + 1/3·9-s + 0.866·12-s − 1.03·15-s + 5/4·16-s + 2.91·17-s − 2.68·20-s + 0.218·21-s + 2/5·25-s − 0.192·27-s + 0.566·28-s − 0.676·35-s − 1/2·36-s + 1.97·37-s − 0.624·41-s − 1.21·43-s + 0.596·45-s − 0.721·48-s + 1/7·49-s − 1.68·51-s − 3.12·59-s + 1.54·60-s − 0.125·63-s − 3/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9261 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9261 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7969059728\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7969059728\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−11.98659319295797721290412051260, −10.77789244508403227045944829665, −10.18331996404873608095625884352, −9.863214598051417092763914791206, −9.457705101580370517739336150901, −9.163045893798789171919999202805, −7.977686193886306753143677629185, −7.81295430367747747098376616892, −6.53484921474790582970192984243, −5.94607665445287604157300008429, −5.48510765590071063887274763802, −5.02709416296405066539370067618, −4.01671863219499060051377207353, −3.05422074105458389777226041971, −1.40786771277750203137263322761,
1.40786771277750203137263322761, 3.05422074105458389777226041971, 4.01671863219499060051377207353, 5.02709416296405066539370067618, 5.48510765590071063887274763802, 5.94607665445287604157300008429, 6.53484921474790582970192984243, 7.81295430367747747098376616892, 7.977686193886306753143677629185, 9.163045893798789171919999202805, 9.457705101580370517739336150901, 9.863214598051417092763914791206, 10.18331996404873608095625884352, 10.77789244508403227045944829665, 11.98659319295797721290412051260