L(s) = 1 | + 2·3-s + 5-s − 3·7-s + 9-s + 3·11-s + 4·13-s + 2·15-s + 5·17-s + 19-s − 6·21-s − 4·25-s − 4·27-s − 2·29-s + 8·31-s + 6·33-s − 3·35-s + 10·37-s + 8·39-s + 6·41-s + 7·43-s + 45-s − 9·47-s + 2·49-s + 10·51-s + 8·53-s + 3·55-s + 2·57-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 0.447·5-s − 1.13·7-s + 1/3·9-s + 0.904·11-s + 1.10·13-s + 0.516·15-s + 1.21·17-s + 0.229·19-s − 1.30·21-s − 4/5·25-s − 0.769·27-s − 0.371·29-s + 1.43·31-s + 1.04·33-s − 0.507·35-s + 1.64·37-s + 1.28·39-s + 0.937·41-s + 1.06·43-s + 0.149·45-s − 1.31·47-s + 2/7·49-s + 1.40·51-s + 1.09·53-s + 0.404·55-s + 0.264·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.577861644\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.577861644\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 15 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.563320344627364655050372899522, −9.075343088620985226404492276671, −8.186562113914066026677579677805, −7.43537131861627444263317650332, −6.20879084669067007942487517486, −5.89904763388088518363904025264, −4.19084046620099313866498183608, −3.41145508666670040670114085640, −2.67622771685970545269516699451, −1.26674131115266326801058270565,
1.26674131115266326801058270565, 2.67622771685970545269516699451, 3.41145508666670040670114085640, 4.19084046620099313866498183608, 5.89904763388088518363904025264, 6.20879084669067007942487517486, 7.43537131861627444263317650332, 8.186562113914066026677579677805, 9.075343088620985226404492276671, 9.563320344627364655050372899522