| L(s) = 1 | + 3·3-s + 5-s + 6·9-s − 11-s − 6·13-s + 3·15-s − 4·17-s + 6·19-s + 3·23-s − 4·25-s + 9·27-s − 4·29-s − 9·31-s − 3·33-s + 7·37-s − 18·39-s − 2·41-s + 6·43-s + 6·45-s + 12·47-s − 7·49-s − 12·51-s + 2·53-s − 55-s + 18·57-s + 9·59-s + 8·61-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.447·5-s + 2·9-s − 0.301·11-s − 1.66·13-s + 0.774·15-s − 0.970·17-s + 1.37·19-s + 0.625·23-s − 4/5·25-s + 1.73·27-s − 0.742·29-s − 1.61·31-s − 0.522·33-s + 1.15·37-s − 2.88·39-s − 0.312·41-s + 0.914·43-s + 0.894·45-s + 1.75·47-s − 49-s − 1.68·51-s + 0.274·53-s − 0.134·55-s + 2.38·57-s + 1.17·59-s + 1.02·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.289347848\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.289347848\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 15 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 5 T + p T^{2} \) |
| 97 | \( 1 + 3 T + p T^{2} \) |
| 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
−11.48765966144200263153367178743, −10.14889382700236447566174943288, −9.451459787186287547513931970345, −8.898169270138754335636599528060, −7.57043482421816953066681680217, −7.26196412622229333930577861479, −5.47596345374224228505212048376, −4.20924970695423611570948148225, −2.91009289331815173285446803923, −2.05183530740595856747996297853,
2.05183530740595856747996297853, 2.91009289331815173285446803923, 4.20924970695423611570948148225, 5.47596345374224228505212048376, 7.26196412622229333930577861479, 7.57043482421816953066681680217, 8.898169270138754335636599528060, 9.451459787186287547513931970345, 10.14889382700236447566174943288, 11.48765966144200263153367178743
