L(s) = 1 | + 2·3-s − 4-s + 3·9-s − 2·12-s + 6·13-s + 16-s + 14·17-s + 2·23-s + 9·25-s + 4·27-s − 2·29-s − 3·36-s + 12·39-s − 10·43-s + 2·48-s − 49-s + 28·51-s − 6·52-s − 12·53-s − 26·61-s − 64-s − 14·68-s + 4·69-s + 18·75-s − 24·79-s + 5·81-s − 4·87-s + ⋯ |
L(s) = 1 | + 1.15·3-s − 1/2·4-s + 9-s − 0.577·12-s + 1.66·13-s + 1/4·16-s + 3.39·17-s + 0.417·23-s + 9/5·25-s + 0.769·27-s − 0.371·29-s − 1/2·36-s + 1.92·39-s − 1.52·43-s + 0.288·48-s − 1/7·49-s + 3.92·51-s − 0.832·52-s − 1.64·53-s − 3.32·61-s − 1/8·64-s − 1.69·68-s + 0.481·69-s + 2.07·75-s − 2.70·79-s + 5/9·81-s − 0.428·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.165817439\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.165817439\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| 13 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
good | 5 | $C_2^2$ | \( 1 - 9 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 29 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 73 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 23 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 162 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 158 T^{2} + p^{2} T^{4} \) |
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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
−10.76478819585551596315768989104, −10.57712839766422400962204337484, −9.909433120286834848958008835898, −9.833323138282610486409897849216, −9.037207096057459242685360680499, −8.912382805672646779752354853771, −8.393621315860024309714270669955, −7.949835094309691188515700603833, −7.61178950661489259753692982695, −7.21011486025012968461358392795, −6.35093675030024605547797105182, −6.08086898664347684779911898670, −5.34465390238234539769941603269, −4.94946932056961628938788272653, −4.27516766227616299875620750773, −3.50561592638930415211628589898, −3.23123754426432929052188179294, −2.95746342164692720974343031467, −1.38390742784676669259892210015, −1.31855408242780957291214467754,
1.31855408242780957291214467754, 1.38390742784676669259892210015, 2.95746342164692720974343031467, 3.23123754426432929052188179294, 3.50561592638930415211628589898, 4.27516766227616299875620750773, 4.94946932056961628938788272653, 5.34465390238234539769941603269, 6.08086898664347684779911898670, 6.35093675030024605547797105182, 7.21011486025012968461358392795, 7.61178950661489259753692982695, 7.949835094309691188515700603833, 8.393621315860024309714270669955, 8.912382805672646779752354853771, 9.037207096057459242685360680499, 9.833323138282610486409897849216, 9.909433120286834848958008835898, 10.57712839766422400962204337484, 10.76478819585551596315768989104