| L(s) = 1 | + 6·3-s + 27·9-s − 40·11-s − 68·17-s − 104·19-s − 242·25-s + 108·27-s − 240·33-s − 52·41-s − 504·43-s − 486·49-s − 408·51-s − 624·57-s − 728·59-s − 1.25e3·67-s + 676·73-s − 1.45e3·75-s + 405·81-s − 2.07e3·83-s + 468·89-s − 356·97-s − 1.08e3·99-s − 2.80e3·107-s + 2.75e3·113-s − 1.46e3·121-s − 312·123-s + 127-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 9-s − 1.09·11-s − 0.970·17-s − 1.25·19-s − 1.93·25-s + 0.769·27-s − 1.26·33-s − 0.198·41-s − 1.78·43-s − 1.41·49-s − 1.12·51-s − 1.45·57-s − 1.60·59-s − 2.29·67-s + 1.08·73-s − 2.23·75-s + 5/9·81-s − 2.74·83-s + 0.557·89-s − 0.372·97-s − 1.09·99-s − 2.53·107-s + 2.29·113-s − 1.09·121-s − 0.228·123-s + 0.000698·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 242 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 486 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 20 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 2826 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 2 p T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 52 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 20462 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 8450 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 47414 T^{2} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 27578 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 26 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 252 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 88574 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 166894 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 364 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 86838 T^{2} + p^{6} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 628 T + p^{3} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 604430 T^{2} + p^{6} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 338 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 363350 T^{2} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 1036 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 234 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 178 T + p^{3} T^{2} )^{2} \) |
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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
−9.589992128455905878883699195050, −9.395679435126117750119942833626, −8.628915338187413241312700958106, −8.541412327293326379377420956127, −8.032473999766586112905357934691, −7.71663202754461129114837029785, −7.24288418682548242680071229224, −6.72083869150646810675559514291, −6.18372567388636763745808723138, −5.87750936112110384139015108259, −4.90633954013428021372520518711, −4.83547845149113053283720499298, −4.00608447879461520111723008154, −3.78282525126594266746677278647, −2.85649507600649371430393740304, −2.70333168971052675208965721674, −1.74630083850338868463303793054, −1.69855989237909698989824609808, 0, 0,
1.69855989237909698989824609808, 1.74630083850338868463303793054, 2.70333168971052675208965721674, 2.85649507600649371430393740304, 3.78282525126594266746677278647, 4.00608447879461520111723008154, 4.83547845149113053283720499298, 4.90633954013428021372520518711, 5.87750936112110384139015108259, 6.18372567388636763745808723138, 6.72083869150646810675559514291, 7.24288418682548242680071229224, 7.71663202754461129114837029785, 8.032473999766586112905357934691, 8.541412327293326379377420956127, 8.628915338187413241312700958106, 9.395679435126117750119942833626, 9.589992128455905878883699195050
