L(s) = 1 | + 8·11-s + 4·17-s − 8·19-s − 2·25-s + 20·41-s − 24·43-s − 6·49-s − 8·59-s − 8·67-s + 4·73-s − 8·83-s + 12·89-s + 28·97-s + 24·107-s − 4·113-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 6·169-s + 173-s + ⋯ |
L(s) = 1 | + 2.41·11-s + 0.970·17-s − 1.83·19-s − 2/5·25-s + 3.12·41-s − 3.65·43-s − 6/7·49-s − 1.04·59-s − 0.977·67-s + 0.468·73-s − 0.878·83-s + 1.27·89-s + 2.84·97-s + 2.32·107-s − 0.376·113-s + 2.36·121-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 + 6/13·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.564528418\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.564528418\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 98 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
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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
−9.010597615319683213644986155059, −8.989863492597320442059366950286, −8.451722972322573378929705484669, −8.156628761998527684171204723400, −7.48963185425083986845197845217, −7.45547387974345659360709584306, −6.68279522622255842988857547982, −6.45366575182029764941814029877, −6.12971458453254814896895988659, −5.96811695028723889743029736212, −5.13812403495844106441880085298, −4.78478903082483479697016840119, −4.19298242447736912347049924654, −4.08982616338089095917752323938, −3.37415196743646865389819279020, −3.23628304153698566465346704262, −2.31841208636882065878583030624, −1.76213458855634401916176121963, −1.39847368315283719129048973238, −0.57162142481473406738212030774,
0.57162142481473406738212030774, 1.39847368315283719129048973238, 1.76213458855634401916176121963, 2.31841208636882065878583030624, 3.23628304153698566465346704262, 3.37415196743646865389819279020, 4.08982616338089095917752323938, 4.19298242447736912347049924654, 4.78478903082483479697016840119, 5.13812403495844106441880085298, 5.96811695028723889743029736212, 6.12971458453254814896895988659, 6.45366575182029764941814029877, 6.68279522622255842988857547982, 7.45547387974345659360709584306, 7.48963185425083986845197845217, 8.156628761998527684171204723400, 8.451722972322573378929705484669, 8.989863492597320442059366950286, 9.010597615319683213644986155059