| L(s) = 1 | − 100·25-s − 142·49-s − 388·73-s − 668·97-s − 484·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 674·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
| L(s) = 1 | − 4·25-s − 2.89·49-s − 5.31·73-s − 6.88·97-s − 4·121-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 − 3.98·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + 0.00448·223-s + 0.00440·227-s + 0.00436·229-s + 0.00429·233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1813347791\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1813347791\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 7 | $C_2^2$ | \( ( 1 + 71 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 - T + p^{2} T^{2} )^{2}( 1 + T + p^{2} T^{2} )^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 601 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 29 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 + 194 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 47 T + p^{2} T^{2} )^{2}( 1 + 47 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 3214 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 53 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 59 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 - 121 T + p^{2} T^{2} )^{2}( 1 + 121 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 2903 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 73 | $C_2$ | \( ( 1 + 97 T + p^{2} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 4679 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 97 | $C_2$ | \( ( 1 + 167 T + p^{2} T^{2} )^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.36316732501943655243870964109, −6.21958724293408002342594003677, −6.10070939398177642624342968401, −5.70577537305413716012082877739, −5.55101890335721145547067257816, −5.45410050542516433914827357127, −5.36897378024111961227105300580, −4.89109681489395960392544966517, −4.52717429702068371643725573501, −4.45763092661184729573160644324, −4.35709884573216801902004522005, −3.89220453466778426556960356743, −3.76783470059572570566200213757, −3.69657950857931745068457954687, −3.34481484479885725709132914028, −2.79634584167771325621804483771, −2.72829059021338353600116407169, −2.70962216530542058294491793837, −2.22282462162336382155923245887, −1.69078622287830428976148504395, −1.57977895742488037898445304379, −1.51387794702189989953977249665, −1.15977788631638934012281273204, −0.28867666947777080170707471485, −0.10374960372011467261098378236,
0.10374960372011467261098378236, 0.28867666947777080170707471485, 1.15977788631638934012281273204, 1.51387794702189989953977249665, 1.57977895742488037898445304379, 1.69078622287830428976148504395, 2.22282462162336382155923245887, 2.70962216530542058294491793837, 2.72829059021338353600116407169, 2.79634584167771325621804483771, 3.34481484479885725709132914028, 3.69657950857931745068457954687, 3.76783470059572570566200213757, 3.89220453466778426556960356743, 4.35709884573216801902004522005, 4.45763092661184729573160644324, 4.52717429702068371643725573501, 4.89109681489395960392544966517, 5.36897378024111961227105300580, 5.45410050542516433914827357127, 5.55101890335721145547067257816, 5.70577537305413716012082877739, 6.10070939398177642624342968401, 6.21958724293408002342594003677, 6.36316732501943655243870964109
