| L(s) = 1 | − 8·4-s + 48·16-s − 52·19-s + 118·31-s + 94·49-s + 148·61-s − 256·64-s + 416·76-s − 22·79-s − 142·109-s + 242·121-s − 944·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 337·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | − 2·4-s + 3·16-s − 2.73·19-s + 3.80·31-s + 1.91·49-s + 2.42·61-s − 4·64-s + 5.47·76-s − 0.278·79-s − 1.30·109-s + 2·121-s − 7.61·124-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 1.99·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 455625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 455625 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.080802689\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.080802689\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 94 T^{2} + p^{4} T^{4} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 337 T^{2} + p^{4} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 26 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 59 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 2591 T^{2} + p^{4} T^{4} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 3191 T^{2} + p^{4} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 74 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 2903 T^{2} + p^{4} T^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 9791 T^{2} + p^{4} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 11 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 18814 T^{2} + p^{4} T^{4} \) |
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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
−10.18546861876357309468658525272, −10.14356040512466379182437624506, −9.749000651502445050068341945387, −9.118428254379663215089952585563, −8.600948506083361128686575591636, −8.508000410327142811075582730748, −8.201335108761875618352897707577, −7.68948344430830276652256711381, −6.86869415871888937036536258756, −6.48464277455651493605613008494, −5.93931158501426928708631355977, −5.53737674180060634657472703949, −4.75238724906576331616573548860, −4.55532207533847170635887582461, −4.12778020716265750665551014916, −3.70142060148537249596650624000, −2.81600718959026266394074723222, −2.23287891833946330367671062167, −1.05461521776962568644347660681, −0.47679738426095166653753509880,
0.47679738426095166653753509880, 1.05461521776962568644347660681, 2.23287891833946330367671062167, 2.81600718959026266394074723222, 3.70142060148537249596650624000, 4.12778020716265750665551014916, 4.55532207533847170635887582461, 4.75238724906576331616573548860, 5.53737674180060634657472703949, 5.93931158501426928708631355977, 6.48464277455651493605613008494, 6.86869415871888937036536258756, 7.68948344430830276652256711381, 8.201335108761875618352897707577, 8.508000410327142811075582730748, 8.600948506083361128686575591636, 9.118428254379663215089952585563, 9.749000651502445050068341945387, 10.14356040512466379182437624506, 10.18546861876357309468658525272
