| L(s) = 1 | − 9-s + 4·11-s + 14·19-s + 4·29-s − 10·31-s + 24·41-s + 5·49-s + 12·59-s − 26·61-s − 8·71-s + 16·79-s + 81-s − 32·89-s − 4·99-s − 18·109-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 17·169-s − 14·171-s + ⋯ |
| L(s) = 1 | − 1/3·9-s + 1.20·11-s + 3.21·19-s + 0.742·29-s − 1.79·31-s + 3.74·41-s + 5/7·49-s + 1.56·59-s − 3.32·61-s − 0.949·71-s + 1.80·79-s + 1/9·81-s − 3.39·89-s − 0.402·99-s − 1.72·109-s − 0.909·121-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 + 1.30·169-s − 1.07·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.189484442\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.189484442\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 77 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 145 T^{2} + p^{2} T^{4} \) |
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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
−10.86017073575357971655165105441, −10.67371455071584097676060671901, −9.706034971630114849990642581933, −9.512403782821587488168576259098, −9.311727942086990365158212016738, −8.870499529610187065756594702120, −8.234180655844527737670026617350, −7.59654940602478680320763469857, −7.39498139352411021883189878660, −7.05756624392090323547003701269, −6.23262787886658877180646550497, −5.89986252985925105185926952266, −5.41116661655413674771725039813, −4.98656065022706237609613720348, −4.07664135888561656889986201682, −3.90616489776559888802834786731, −3.00828295337730168631484694991, −2.72428241094245306777605376979, −1.52137860185798936535033025746, −0.942396002529228916396980195111,
0.942396002529228916396980195111, 1.52137860185798936535033025746, 2.72428241094245306777605376979, 3.00828295337730168631484694991, 3.90616489776559888802834786731, 4.07664135888561656889986201682, 4.98656065022706237609613720348, 5.41116661655413674771725039813, 5.89986252985925105185926952266, 6.23262787886658877180646550497, 7.05756624392090323547003701269, 7.39498139352411021883189878660, 7.59654940602478680320763469857, 8.234180655844527737670026617350, 8.870499529610187065756594702120, 9.311727942086990365158212016738, 9.512403782821587488168576259098, 9.706034971630114849990642581933, 10.67371455071584097676060671901, 10.86017073575357971655165105441
