| L(s) = 1 | + 5-s − 2·7-s − 3·9-s + 4·11-s + 4·13-s + 6·17-s − 19-s − 2·23-s + 25-s + 6·29-s − 8·31-s − 2·35-s − 4·37-s + 6·41-s + 6·43-s − 3·45-s + 6·47-s − 3·49-s − 8·53-s + 4·55-s + 12·59-s − 6·61-s + 6·63-s + 4·65-s − 10·73-s − 8·77-s − 8·79-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.755·7-s − 9-s + 1.20·11-s + 1.10·13-s + 1.45·17-s − 0.229·19-s − 0.417·23-s + 1/5·25-s + 1.11·29-s − 1.43·31-s − 0.338·35-s − 0.657·37-s + 0.937·41-s + 0.914·43-s − 0.447·45-s + 0.875·47-s − 3/7·49-s − 1.09·53-s + 0.539·55-s + 1.56·59-s − 0.768·61-s + 0.755·63-s + 0.496·65-s − 1.17·73-s − 0.911·77-s − 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.119490542\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.119490542\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−8.177049615497252411299907897505, −7.30074003333878305067600101824, −6.47045167529456579797345488883, −5.93417433999626479704340055439, −5.53249748476365579405431705816, −4.29237908609109024161935375003, −3.50601854714300152531701253949, −2.97447858271543835452076492685, −1.77125895929780636050891540531, −0.78696996676608591793609930402,
0.78696996676608591793609930402, 1.77125895929780636050891540531, 2.97447858271543835452076492685, 3.50601854714300152531701253949, 4.29237908609109024161935375003, 5.53249748476365579405431705816, 5.93417433999626479704340055439, 6.47045167529456579797345488883, 7.30074003333878305067600101824, 8.177049615497252411299907897505
