| L(s) = 1 | − 8.74e3·9-s + 1.56e5·25-s − 3.14e6·41-s + 1.35e5·49-s + 4.78e7·81-s − 2.22e7·89-s − 3.21e7·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2.50e8·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s − 1.36e9·225-s + ⋯ |
| L(s) = 1 | − 4·9-s + 2·25-s − 7.12·41-s + 0.164·49-s + 10·81-s − 3.34·89-s − 1.64·121-s + 3.99·169-s − 8·225-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.286170341\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.286170341\) |
| \(L(\frac{9}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{4} \) |
| 7 | $C_2^2$ | \( ( 1 - 67866 T^{2} + p^{14} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 16052382 T^{2} + p^{14} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 125276586 T^{2} + p^{14} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 + 644127118 T^{2} + p^{14} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 5951338874 T^{2} + p^{14} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + 100482630246 T^{2} + p^{14} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 785678 T + p^{7} T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 - 985929843946 T^{2} + p^{14} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 1801585778906 T^{2} + p^{14} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 4687670223678 T^{2} + p^{14} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{4} \) |
| 67 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{4} \) |
| 71 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{4} \) |
| 83 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{4} \) |
| 89 | $C_2$ | \( ( 1 + 5555326 T + p^{7} T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.99504002315586086910182328528, −6.76619844931824271488455912023, −6.65783333095068646593132216372, −6.61884373366023727431217462647, −6.12643204560607334572835916700, −5.87042252399153087508852301942, −5.53166205550532669311467210620, −5.34440632842922763943852800048, −5.16234661979750474018784568673, −5.06744762513100003775522164244, −4.73441521012024001379411172438, −4.31513762131835067172579070105, −3.77636218871025502021398589263, −3.55156607759957077082853033169, −3.24817607294262247744771202734, −3.02235106889696695525551007565, −2.95258661313545921437973388717, −2.61888394960306597811395086185, −2.26296753974935737167214988397, −1.82029986949210465619845785938, −1.55119057585529221507785747979, −1.27898768449959581055358212371, −0.49634518120001175770639685067, −0.44822192373776133623264798537, −0.23134152062819957157500628079,
0.23134152062819957157500628079, 0.44822192373776133623264798537, 0.49634518120001175770639685067, 1.27898768449959581055358212371, 1.55119057585529221507785747979, 1.82029986949210465619845785938, 2.26296753974935737167214988397, 2.61888394960306597811395086185, 2.95258661313545921437973388717, 3.02235106889696695525551007565, 3.24817607294262247744771202734, 3.55156607759957077082853033169, 3.77636218871025502021398589263, 4.31513762131835067172579070105, 4.73441521012024001379411172438, 5.06744762513100003775522164244, 5.16234661979750474018784568673, 5.34440632842922763943852800048, 5.53166205550532669311467210620, 5.87042252399153087508852301942, 6.12643204560607334572835916700, 6.61884373366023727431217462647, 6.65783333095068646593132216372, 6.76619844931824271488455912023, 6.99504002315586086910182328528
