| L(s) = 1 | + 2.66·3-s + 3.66·7-s + 4.12·9-s + 1.21·11-s − 2.21·13-s − 1.21·17-s + 2.57·19-s + 9.79·21-s + 23-s + 3.00·27-s + 1.45·29-s + 6.46·31-s + 3.24·33-s − 4·37-s − 5.90·39-s − 10.9·41-s + 6.90·43-s + 5.45·47-s + 6.46·49-s − 3.24·51-s − 3.81·53-s + 6.88·57-s + 4.24·59-s + 6.78·61-s + 15.1·63-s + 12.8·67-s + 2.66·69-s + ⋯ |
| L(s) = 1 | + 1.54·3-s + 1.38·7-s + 1.37·9-s + 0.366·11-s − 0.614·13-s − 0.294·17-s + 0.591·19-s + 2.13·21-s + 0.208·23-s + 0.577·27-s + 0.270·29-s + 1.16·31-s + 0.564·33-s − 0.657·37-s − 0.946·39-s − 1.70·41-s + 1.05·43-s + 0.795·47-s + 0.923·49-s − 0.453·51-s − 0.524·53-s + 0.911·57-s + 0.553·59-s + 0.868·61-s + 1.90·63-s + 1.57·67-s + 0.321·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.867363297\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.867363297\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 - 2.66T + 3T^{2} \) |
| 7 | \( 1 - 3.66T + 7T^{2} \) |
| 11 | \( 1 - 1.21T + 11T^{2} \) |
| 13 | \( 1 + 2.21T + 13T^{2} \) |
| 17 | \( 1 + 1.21T + 17T^{2} \) |
| 19 | \( 1 - 2.57T + 19T^{2} \) |
| 29 | \( 1 - 1.45T + 29T^{2} \) |
| 31 | \( 1 - 6.46T + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 + 10.9T + 41T^{2} \) |
| 43 | \( 1 - 6.90T + 43T^{2} \) |
| 47 | \( 1 - 5.45T + 47T^{2} \) |
| 53 | \( 1 + 3.81T + 53T^{2} \) |
| 59 | \( 1 - 4.24T + 59T^{2} \) |
| 61 | \( 1 - 6.78T + 61T^{2} \) |
| 67 | \( 1 - 12.8T + 67T^{2} \) |
| 71 | \( 1 - 6.91T + 71T^{2} \) |
| 73 | \( 1 - 15.2T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 15.5T + 83T^{2} \) |
| 89 | \( 1 + 10.9T + 89T^{2} \) |
| 97 | \( 1 + 1.69T + 97T^{2} \) |
| 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
−7.85293333407145050838573339124, −7.25577876096084457400967041400, −6.64492165793880973786699875959, −5.44154515635735693908868874486, −4.83302161701574254295754475120, −4.13194975785443966687185147526, −3.39222839745037433545792532768, −2.52188593780110912146938476926, −1.95469104466812372368314169219, −1.04642958277531207704167110340,
1.04642958277531207704167110340, 1.95469104466812372368314169219, 2.52188593780110912146938476926, 3.39222839745037433545792532768, 4.13194975785443966687185147526, 4.83302161701574254295754475120, 5.44154515635735693908868874486, 6.64492165793880973786699875959, 7.25577876096084457400967041400, 7.85293333407145050838573339124
