L(s) = 1 | + 3-s + 1.04·5-s + 1.46·7-s + 9-s − 2.04·11-s − 6.89·13-s + 1.04·15-s − 7.77·17-s + 6.37·19-s + 1.46·21-s − 3.90·25-s + 27-s + 4.14·29-s + 0.909·31-s − 2.04·33-s + 1.53·35-s + 4.07·37-s − 6.89·39-s + 4.89·41-s − 3.50·43-s + 1.04·45-s − 1.92·47-s − 4.85·49-s − 7.77·51-s − 7.82·53-s − 2.14·55-s + 6.37·57-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.468·5-s + 0.552·7-s + 0.333·9-s − 0.617·11-s − 1.91·13-s + 0.270·15-s − 1.88·17-s + 1.46·19-s + 0.319·21-s − 0.780·25-s + 0.192·27-s + 0.770·29-s + 0.163·31-s − 0.356·33-s + 0.258·35-s + 0.670·37-s − 1.10·39-s + 0.764·41-s − 0.534·43-s + 0.156·45-s − 0.281·47-s − 0.694·49-s − 1.08·51-s − 1.07·53-s − 0.289·55-s + 0.844·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6348 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6348 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 23 | \( 1 \) |
good | 5 | \( 1 - 1.04T + 5T^{2} \) |
| 7 | \( 1 - 1.46T + 7T^{2} \) |
| 11 | \( 1 + 2.04T + 11T^{2} \) |
| 13 | \( 1 + 6.89T + 13T^{2} \) |
| 17 | \( 1 + 7.77T + 17T^{2} \) |
| 19 | \( 1 - 6.37T + 19T^{2} \) |
| 29 | \( 1 - 4.14T + 29T^{2} \) |
| 31 | \( 1 - 0.909T + 31T^{2} \) |
| 37 | \( 1 - 4.07T + 37T^{2} \) |
| 41 | \( 1 - 4.89T + 41T^{2} \) |
| 43 | \( 1 + 3.50T + 43T^{2} \) |
| 47 | \( 1 + 1.92T + 47T^{2} \) |
| 53 | \( 1 + 7.82T + 53T^{2} \) |
| 59 | \( 1 - 14.4T + 59T^{2} \) |
| 61 | \( 1 + 7.53T + 61T^{2} \) |
| 67 | \( 1 + 10.0T + 67T^{2} \) |
| 71 | \( 1 + 0.197T + 71T^{2} \) |
| 73 | \( 1 + 7.96T + 73T^{2} \) |
| 79 | \( 1 + 4.62T + 79T^{2} \) |
| 83 | \( 1 - 8.28T + 83T^{2} \) |
| 89 | \( 1 + 0.275T + 89T^{2} \) |
| 97 | \( 1 + 7.35T + 97T^{2} \) |
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
−7.71214918718823883568050248058, −7.13075249929824912666213514764, −6.37013122401533790140969658246, −5.33746463113187742987173644371, −4.82149967159521450054225825227, −4.17414846076972141810006949441, −2.83676135520376319895738740604, −2.46925220966909041090201321314, −1.54845082355620012018223062559, 0,
1.54845082355620012018223062559, 2.46925220966909041090201321314, 2.83676135520376319895738740604, 4.17414846076972141810006949441, 4.82149967159521450054225825227, 5.33746463113187742987173644371, 6.37013122401533790140969658246, 7.13075249929824912666213514764, 7.71214918718823883568050248058