L(s) = 1 | + 3-s − 3.70·5-s − 1.62·7-s + 9-s − 4.14·11-s − 5.66·13-s − 3.70·15-s + 0.274·17-s − 7.65·19-s − 1.62·21-s − 23-s + 8.71·25-s + 27-s + 29-s − 9.73·31-s − 4.14·33-s + 5.99·35-s − 1.32·37-s − 5.66·39-s − 8.38·41-s + 11.4·43-s − 3.70·45-s − 7.35·47-s − 4.37·49-s + 0.274·51-s − 0.0615·53-s + 15.3·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 1.65·5-s − 0.612·7-s + 0.333·9-s − 1.25·11-s − 1.57·13-s − 0.956·15-s + 0.0666·17-s − 1.75·19-s − 0.353·21-s − 0.208·23-s + 1.74·25-s + 0.192·27-s + 0.185·29-s − 1.74·31-s − 0.721·33-s + 1.01·35-s − 0.218·37-s − 0.906·39-s − 1.30·41-s + 1.74·43-s − 0.552·45-s − 1.07·47-s − 0.625·49-s + 0.0384·51-s − 0.00845·53-s + 2.07·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1149480375\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1149480375\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 29 | \( 1 - T \) |
good | 5 | \( 1 + 3.70T + 5T^{2} \) |
| 7 | \( 1 + 1.62T + 7T^{2} \) |
| 11 | \( 1 + 4.14T + 11T^{2} \) |
| 13 | \( 1 + 5.66T + 13T^{2} \) |
| 17 | \( 1 - 0.274T + 17T^{2} \) |
| 19 | \( 1 + 7.65T + 19T^{2} \) |
| 31 | \( 1 + 9.73T + 31T^{2} \) |
| 37 | \( 1 + 1.32T + 37T^{2} \) |
| 41 | \( 1 + 8.38T + 41T^{2} \) |
| 43 | \( 1 - 11.4T + 43T^{2} \) |
| 47 | \( 1 + 7.35T + 47T^{2} \) |
| 53 | \( 1 + 0.0615T + 53T^{2} \) |
| 59 | \( 1 + 1.61T + 59T^{2} \) |
| 61 | \( 1 - 8.30T + 61T^{2} \) |
| 67 | \( 1 - 1.88T + 67T^{2} \) |
| 71 | \( 1 + 12.7T + 71T^{2} \) |
| 73 | \( 1 + 1.88T + 73T^{2} \) |
| 79 | \( 1 + 13.8T + 79T^{2} \) |
| 83 | \( 1 + 10.8T + 83T^{2} \) |
| 89 | \( 1 - 13.3T + 89T^{2} \) |
| 97 | \( 1 + 5.34T + 97T^{2} \) |
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
−7.80577017499107927215973935617, −7.28113596602380580738115156246, −6.80793299194211442008601587355, −5.67941227771329127500172755167, −4.77817341759307409270921303847, −4.27043687146300992672786918470, −3.45954000842375492754365968938, −2.80459146215001240769540596489, −2.02077341141627581970612110557, −0.14939959031164832794127129155,
0.14939959031164832794127129155, 2.02077341141627581970612110557, 2.80459146215001240769540596489, 3.45954000842375492754365968938, 4.27043687146300992672786918470, 4.77817341759307409270921303847, 5.67941227771329127500172755167, 6.80793299194211442008601587355, 7.28113596602380580738115156246, 7.80577017499107927215973935617