| L(s) = 1 | + 16·25-s + 28·49-s − 64·73-s − 32·97-s + 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 20·169-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 | + 16/5·25-s + 4·49-s − 7.49·73-s − 3.24·97-s + 4·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.53·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.516127581\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.516127581\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 16 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 + 40 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 80 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 - 56 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 89 | $C_2^2$ | \( ( 1 - 160 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
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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
−6.51583295633916456105894335849, −6.15427154776584537075707952763, −5.94281318274246497042621590415, −5.65087904591966964672412523318, −5.61225846447941715209799195022, −5.44696298790095134222291905347, −5.14541796010183028049516563376, −5.04255642342924243220267708301, −4.45228191969632204645319055120, −4.40398633882449808527785892196, −4.39343322099944335165981441256, −4.17354305510415569687978755613, −4.00392677020688365232810016452, −3.43198511704205012337449992451, −3.21554199202807981722130805072, −2.93554996838928646773357707858, −2.93329519775264030072175111275, −2.82908884832523445124157183624, −2.19232052052139528769633708207, −2.15102708436177572238775636195, −1.78661411746281092730120268500, −1.20716232499626113037519851034, −1.09438050715661994378433498138, −0.945711542954576152791818881692, −0.20421703015655479918572033501,
0.20421703015655479918572033501, 0.945711542954576152791818881692, 1.09438050715661994378433498138, 1.20716232499626113037519851034, 1.78661411746281092730120268500, 2.15102708436177572238775636195, 2.19232052052139528769633708207, 2.82908884832523445124157183624, 2.93329519775264030072175111275, 2.93554996838928646773357707858, 3.21554199202807981722130805072, 3.43198511704205012337449992451, 4.00392677020688365232810016452, 4.17354305510415569687978755613, 4.39343322099944335165981441256, 4.40398633882449808527785892196, 4.45228191969632204645319055120, 5.04255642342924243220267708301, 5.14541796010183028049516563376, 5.44696298790095134222291905347, 5.61225846447941715209799195022, 5.65087904591966964672412523318, 5.94281318274246497042621590415, 6.15427154776584537075707952763, 6.51583295633916456105894335849
