| L(s) = 1 | + 24·23-s − 8·49-s − 8·61-s − 48·83-s − 48·107-s + 8·109-s − 32·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 40·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯ |
| L(s) = 1 | + 5.00·23-s − 8/7·49-s − 1.02·61-s − 5.26·83-s − 4.64·107-s + 0.766·109-s − 2.90·121-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 − 3.07·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.338634220\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.338634220\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 16 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 68 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 32 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 - 32 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 + 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 122 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{4} \) |
| 89 | $C_2^2$ | \( ( 1 - 128 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−5.90864701183165296836443511903, −5.86046456979622832516022039415, −5.59054689623734520625430356669, −5.36282107691616203200238716255, −5.17066798793285467707879429661, −5.14245896247348357452354047431, −4.98624054844034574618216762886, −4.59578124577727011142463794156, −4.40288220707656528753536885975, −4.33121724823998600838461356250, −3.98566438696548103017336773671, −3.92596931410326313366306004771, −3.44703892837383429461764689647, −3.17363352263684097652290955903, −3.08670799525767381793137261540, −2.99351894258413739902040780091, −2.62476890985688764657500164813, −2.61383947023372432345368669482, −2.27099012751212271749050565260, −1.64368827629635899890952921285, −1.55762696929710676715380007907, −1.27381886685854053193989957683, −1.13467550804488803505607132453, −0.72955349109417211946759886466, −0.16714803001958414373733052960,
0.16714803001958414373733052960, 0.72955349109417211946759886466, 1.13467550804488803505607132453, 1.27381886685854053193989957683, 1.55762696929710676715380007907, 1.64368827629635899890952921285, 2.27099012751212271749050565260, 2.61383947023372432345368669482, 2.62476890985688764657500164813, 2.99351894258413739902040780091, 3.08670799525767381793137261540, 3.17363352263684097652290955903, 3.44703892837383429461764689647, 3.92596931410326313366306004771, 3.98566438696548103017336773671, 4.33121724823998600838461356250, 4.40288220707656528753536885975, 4.59578124577727011142463794156, 4.98624054844034574618216762886, 5.14245896247348357452354047431, 5.17066798793285467707879429661, 5.36282107691616203200238716255, 5.59054689623734520625430356669, 5.86046456979622832516022039415, 5.90864701183165296836443511903
