| L(s) = 1 | − 2·3-s − 3·9-s − 6·11-s + 6·17-s + 2·19-s + 14·27-s + 12·33-s + 18·41-s − 8·43-s − 2·49-s − 12·51-s − 4·57-s + 24·59-s − 22·67-s − 14·73-s − 4·81-s − 30·83-s + 6·89-s + 28·97-s + 18·99-s − 18·107-s − 30·113-s + 5·121-s − 36·123-s + 127-s + 16·129-s + 131-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 9-s − 1.80·11-s + 1.45·17-s + 0.458·19-s + 2.69·27-s + 2.08·33-s + 2.81·41-s − 1.21·43-s − 2/7·49-s − 1.68·51-s − 0.529·57-s + 3.12·59-s − 2.68·67-s − 1.63·73-s − 4/9·81-s − 3.29·83-s + 0.635·89-s + 2.84·97-s + 1.80·99-s − 1.74·107-s − 2.82·113-s + 5/11·121-s − 3.24·123-s + 0.0887·127-s + 1.40·129-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 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
−7.78780378961370343832532905981, −7.59730464086360264715983843332, −7.12801620229096879882879342161, −6.83242231925202286569576968901, −6.17662238146902434575253380162, −5.98413333545002015693438473382, −5.62920368284516816425504237223, −5.52626339177379673315659638160, −5.04318839647285545666921392296, −4.90624736158526718878273568931, −4.29540563998387000260642623835, −3.86729398559344607618242963697, −3.18710741915481272596207289518, −3.02128450589521995989985113229, −2.52033648665040371102930659032, −2.34186024121928306578464351097, −1.23166933464039151195080706374, −1.02851184903277241635924384639, 0, 0,
1.02851184903277241635924384639, 1.23166933464039151195080706374, 2.34186024121928306578464351097, 2.52033648665040371102930659032, 3.02128450589521995989985113229, 3.18710741915481272596207289518, 3.86729398559344607618242963697, 4.29540563998387000260642623835, 4.90624736158526718878273568931, 5.04318839647285545666921392296, 5.52626339177379673315659638160, 5.62920368284516816425504237223, 5.98413333545002015693438473382, 6.17662238146902434575253380162, 6.83242231925202286569576968901, 7.12801620229096879882879342161, 7.59730464086360264715983843332, 7.78780378961370343832532905981
