| L(s) = 1 | − 2·3-s + 16·5-s + 7·7-s − 23·9-s + 24·11-s + 68·13-s − 32·15-s + 54·17-s − 46·19-s − 14·21-s − 176·23-s + 131·25-s + 100·27-s + 174·29-s + 116·31-s − 48·33-s + 112·35-s − 74·37-s − 136·39-s − 10·41-s − 480·43-s − 368·45-s + 572·47-s + 49·49-s − 108·51-s + 162·53-s + 384·55-s + ⋯ |
| L(s) = 1 | − 0.384·3-s + 1.43·5-s + 0.377·7-s − 0.851·9-s + 0.657·11-s + 1.45·13-s − 0.550·15-s + 0.770·17-s − 0.555·19-s − 0.145·21-s − 1.59·23-s + 1.04·25-s + 0.712·27-s + 1.11·29-s + 0.672·31-s − 0.253·33-s + 0.540·35-s − 0.328·37-s − 0.558·39-s − 0.0380·41-s − 1.70·43-s − 1.21·45-s + 1.77·47-s + 1/7·49-s − 0.296·51-s + 0.419·53-s + 0.941·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.404194952\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.404194952\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - p T \) |
| good | 3 | \( 1 + 2 T + p^{3} T^{2} \) |
| 5 | \( 1 - 16 T + p^{3} T^{2} \) |
| 11 | \( 1 - 24 T + p^{3} T^{2} \) |
| 13 | \( 1 - 68 T + p^{3} T^{2} \) |
| 17 | \( 1 - 54 T + p^{3} T^{2} \) |
| 19 | \( 1 + 46 T + p^{3} T^{2} \) |
| 23 | \( 1 + 176 T + p^{3} T^{2} \) |
| 29 | \( 1 - 6 p T + p^{3} T^{2} \) |
| 31 | \( 1 - 116 T + p^{3} T^{2} \) |
| 37 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 41 | \( 1 + 10 T + p^{3} T^{2} \) |
| 43 | \( 1 + 480 T + p^{3} T^{2} \) |
| 47 | \( 1 - 572 T + p^{3} T^{2} \) |
| 53 | \( 1 - 162 T + p^{3} T^{2} \) |
| 59 | \( 1 + 86 T + p^{3} T^{2} \) |
| 61 | \( 1 - 904 T + p^{3} T^{2} \) |
| 67 | \( 1 - 660 T + p^{3} T^{2} \) |
| 71 | \( 1 + 1024 T + p^{3} T^{2} \) |
| 73 | \( 1 - 770 T + p^{3} T^{2} \) |
| 79 | \( 1 - 904 T + p^{3} T^{2} \) |
| 83 | \( 1 - 682 T + p^{3} T^{2} \) |
| 89 | \( 1 + 102 T + p^{3} T^{2} \) |
| 97 | \( 1 + 218 T + p^{3} T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.56480593280988231971019898375, −9.945878943104851608220139975777, −8.813979355129286989845266630100, −8.216718257510508179639488871985, −6.49918233833926407681722809876, −6.05012059571621196865816067982, −5.20110970311538089375129196469, −3.75046211898536870871056101514, −2.26641089135339627238958984135, −1.08599772617165914430570884843,
1.08599772617165914430570884843, 2.26641089135339627238958984135, 3.75046211898536870871056101514, 5.20110970311538089375129196469, 6.05012059571621196865816067982, 6.49918233833926407681722809876, 8.216718257510508179639488871985, 8.813979355129286989845266630100, 9.945878943104851608220139975777, 10.56480593280988231971019898375
