L(s) = 1 | − 52·9-s − 52·11-s − 296·23-s − 242·25-s − 236·29-s − 508·37-s + 244·43-s − 340·53-s + 840·67-s + 840·71-s + 2.10e3·79-s + 1.97e3·81-s + 2.70e3·99-s + 1.76e3·107-s + 1.19e3·109-s − 3.09e3·113-s − 634·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 3.24e3·169-s + ⋯ |
L(s) = 1 | − 1.92·9-s − 1.42·11-s − 2.68·23-s − 1.93·25-s − 1.51·29-s − 2.25·37-s + 0.865·43-s − 0.881·53-s + 1.53·67-s + 1.40·71-s + 2.99·79-s + 2.70·81-s + 2.74·99-s + 1.59·107-s + 1.05·109-s − 2.57·113-s − 0.476·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s − 1.47·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38416 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
good | 3 | $C_2^2$ | \( 1 + 52 T^{2} + p^{6} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 242 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 26 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 3242 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 832 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 4740 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 148 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 118 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 28618 T^{2} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 254 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 129392 T^{2} + p^{6} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 122 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 112598 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 170 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 318308 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 84162 T^{2} + p^{6} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 420 T + p^{3} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 420 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 116784 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 1052 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 957676 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 358688 T^{2} + p^{6} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 1725888 T^{2} + p^{6} T^{4} \) |
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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
−11.71712420609773471690665250998, −11.39060317175056551334154486048, −10.79267688741083356672135526705, −10.47487367177661302017475641176, −9.655876193632681288055235369055, −9.528622967899325740229695544400, −8.577149791689350606159057502039, −8.277507422016815221742895073099, −7.77066322685964034313537065424, −7.46661262613619143887952057991, −6.22022877222351440071548759067, −6.09514041788396441401649172021, −5.31995382662879687573706734978, −5.13323057035691803607918386971, −3.71782982636151383795871771327, −3.62332699959918104552027039313, −2.29525485551279358489498593280, −2.14463166975265245849982783769, 0, 0,
2.14463166975265245849982783769, 2.29525485551279358489498593280, 3.62332699959918104552027039313, 3.71782982636151383795871771327, 5.13323057035691803607918386971, 5.31995382662879687573706734978, 6.09514041788396441401649172021, 6.22022877222351440071548759067, 7.46661262613619143887952057991, 7.77066322685964034313537065424, 8.277507422016815221742895073099, 8.577149791689350606159057502039, 9.528622967899325740229695544400, 9.655876193632681288055235369055, 10.47487367177661302017475641176, 10.79267688741083356672135526705, 11.39060317175056551334154486048, 11.71712420609773471690665250998