| L(s) = 1 | − 3-s + 4-s + 9-s − 12-s + 12·13-s + 16-s − 8·19-s + 25-s − 27-s + 36-s + 12·37-s − 12·39-s + 8·43-s − 48-s − 14·49-s + 12·52-s + 8·57-s − 4·61-s + 64-s − 8·67-s + 4·73-s − 75-s − 8·76-s − 16·79-s + 81-s + 36·97-s + 100-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/2·4-s + 1/3·9-s − 0.288·12-s + 3.32·13-s + 1/4·16-s − 1.83·19-s + 1/5·25-s − 0.192·27-s + 1/6·36-s + 1.97·37-s − 1.92·39-s + 1.21·43-s − 0.144·48-s − 2·49-s + 1.66·52-s + 1.05·57-s − 0.512·61-s + 1/8·64-s − 0.977·67-s + 0.468·73-s − 0.115·75-s − 0.917·76-s − 1.80·79-s + 1/9·81-s + 3.65·97-s + 1/10·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 326700 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 326700 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.985871164\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.985871164\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_1$ | \( 1 + T \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
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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
−8.806615184716062235501383813449, −8.513019122943059313204005851301, −7.66390458297268966013763020839, −7.57573281855054290442967607898, −6.60621560114120530798939193668, −6.26970322751556902553424782587, −6.00274455214544718923999184915, −5.88065102738235433506263724139, −4.61703453242534826715971634086, −4.54831431199383754044494926069, −3.61918145379138433006007908471, −3.43907663083633253170978783027, −2.39738469075182811951844975438, −1.62793361398186118811552584678, −0.910261434468509007715876124552,
0.910261434468509007715876124552, 1.62793361398186118811552584678, 2.39738469075182811951844975438, 3.43907663083633253170978783027, 3.61918145379138433006007908471, 4.54831431199383754044494926069, 4.61703453242534826715971634086, 5.88065102738235433506263724139, 6.00274455214544718923999184915, 6.26970322751556902553424782587, 6.60621560114120530798939193668, 7.57573281855054290442967607898, 7.66390458297268966013763020839, 8.513019122943059313204005851301, 8.806615184716062235501383813449
