| L(s) = 1 | − 2·3-s − 5·5-s + 28·7-s + 27·9-s − 45·11-s + 118·13-s + 10·15-s + 54·17-s − 121·19-s − 56·21-s + 69·23-s − 154·27-s − 324·29-s − 88·31-s + 90·33-s − 140·35-s + 259·37-s − 236·39-s + 390·41-s + 572·43-s − 135·45-s + 45·47-s + 441·49-s − 108·51-s − 597·53-s + 225·55-s + 242·57-s + ⋯ |
| L(s) = 1 | − 0.384·3-s − 0.447·5-s + 1.51·7-s + 9-s − 1.23·11-s + 2.51·13-s + 0.172·15-s + 0.770·17-s − 1.46·19-s − 0.581·21-s + 0.625·23-s − 1.09·27-s − 2.07·29-s − 0.509·31-s + 0.474·33-s − 0.676·35-s + 1.15·37-s − 0.968·39-s + 1.48·41-s + 2.02·43-s − 0.447·45-s + 0.139·47-s + 9/7·49-s − 0.296·51-s − 1.54·53-s + 0.551·55-s + 0.562·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 313600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 313600 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.776111258\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.776111258\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 7 | $C_2$ | \( 1 - 4 p T + p^{3} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 2 T - 23 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 45 T + 694 T^{2} + 45 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 59 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 54 T - 1997 T^{2} - 54 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 121 T + 7782 T^{2} + 121 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 3 p T - 14 p^{2} T^{2} - 3 p^{4} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 162 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 88 T - 22047 T^{2} + 88 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 7 p T + 12 p^{2} T^{2} - 7 p^{4} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 195 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 286 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 45 T - 101798 T^{2} - 45 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 597 T + 207532 T^{2} + 597 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 360 T - 75779 T^{2} + 360 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 392 T - 73317 T^{2} + 392 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 280 T - 222363 T^{2} + 280 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 48 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 668 T + 57207 T^{2} + 668 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 782 T + 118485 T^{2} - 782 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 768 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 1194 T + 720667 T^{2} - 1194 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 902 T + p^{3} T^{2} )^{2} \) |
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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
−10.91796874532800871770764223549, −10.48363156053982916687705063038, −9.654501098455661572097993301956, −9.114871375564382126972698385374, −8.910069059219240319585256136259, −8.118153379539610668149114124838, −7.927833583667584658875731710814, −7.56875381283536026398297967529, −7.22540348026008709696611574359, −6.30667999154460271280950449961, −5.77707550062993724905007182985, −5.76647327660066276511160302097, −4.92662207633951749705784223142, −4.27556526153179904551614671538, −4.13175858058002997581577793331, −3.48954687785082697490935015396, −2.60924418659676248201970716497, −1.71456253059332330903925781118, −1.39978659894588040544829320853, −0.54509718477450021843362418057,
0.54509718477450021843362418057, 1.39978659894588040544829320853, 1.71456253059332330903925781118, 2.60924418659676248201970716497, 3.48954687785082697490935015396, 4.13175858058002997581577793331, 4.27556526153179904551614671538, 4.92662207633951749705784223142, 5.76647327660066276511160302097, 5.77707550062993724905007182985, 6.30667999154460271280950449961, 7.22540348026008709696611574359, 7.56875381283536026398297967529, 7.927833583667584658875731710814, 8.118153379539610668149114124838, 8.910069059219240319585256136259, 9.114871375564382126972698385374, 9.654501098455661572097993301956, 10.48363156053982916687705063038, 10.91796874532800871770764223549
