| L(s) = 1 | + 8·11-s − 2·25-s − 4·29-s + 20·37-s − 8·43-s − 12·53-s + 24·67-s + 16·79-s + 24·107-s + 4·109-s − 20·113-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 18·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 2.41·11-s − 2/5·25-s − 0.742·29-s + 3.28·37-s − 1.21·43-s − 1.64·53-s + 2.93·67-s + 1.80·79-s + 2.32·107-s + 0.383·109-s − 1.88·113-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 − 1.38·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.657910006\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.657910006\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 162 T^{2} + p^{2} T^{4} \) |
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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
−8.779091007299087085196726220357, −8.337007835671318017318751643484, −7.916360013489859737130574623807, −7.88493473485601880256223068312, −7.14125569437055909063110562296, −6.85057443497995944470937656473, −6.51021406766901593296104689264, −6.24954272416215348011693505943, −5.82589950964702926081659478904, −5.49554046437028619715605872656, −4.71561811968611740316850013120, −4.62859880376613682627643482851, −4.00417021186945233057196570413, −3.81952679683220887622927220385, −3.30970723960901858405009413790, −2.88551589580167821469342231874, −1.96485951166770136167386182802, −1.94223960823426379351982495343, −1.06134693259088244155334188862, −0.67692514804931885144238917639,
0.67692514804931885144238917639, 1.06134693259088244155334188862, 1.94223960823426379351982495343, 1.96485951166770136167386182802, 2.88551589580167821469342231874, 3.30970723960901858405009413790, 3.81952679683220887622927220385, 4.00417021186945233057196570413, 4.62859880376613682627643482851, 4.71561811968611740316850013120, 5.49554046437028619715605872656, 5.82589950964702926081659478904, 6.24954272416215348011693505943, 6.51021406766901593296104689264, 6.85057443497995944470937656473, 7.14125569437055909063110562296, 7.88493473485601880256223068312, 7.916360013489859737130574623807, 8.337007835671318017318751643484, 8.779091007299087085196726220357
