| L(s) = 1 | − 16.3·3-s + 49·7-s + 23.0·9-s + 728.·11-s + 305.·13-s + 490.·17-s − 1.10e3·19-s − 799.·21-s + 3.19e3·23-s + 3.58e3·27-s + 3.73e3·29-s − 3.67e3·31-s − 1.18e4·33-s + 1.57e4·37-s − 4.98e3·39-s − 2.23e3·41-s − 1.40e4·43-s − 2.15e4·47-s + 2.40e3·49-s − 7.99e3·51-s − 1.73e4·53-s + 1.80e4·57-s − 2.66e4·59-s − 2.80e4·61-s + 1.12e3·63-s + 1.89e4·67-s − 5.21e4·69-s + ⋯ |
| L(s) = 1 | − 1.04·3-s + 0.377·7-s + 0.0948·9-s + 1.81·11-s + 0.502·13-s + 0.411·17-s − 0.702·19-s − 0.395·21-s + 1.25·23-s + 0.947·27-s + 0.823·29-s − 0.686·31-s − 1.90·33-s + 1.89·37-s − 0.525·39-s − 0.208·41-s − 1.15·43-s − 1.42·47-s + 0.142·49-s − 0.430·51-s − 0.849·53-s + 0.734·57-s − 0.995·59-s − 0.963·61-s + 0.0358·63-s + 0.514·67-s − 1.31·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.797504719\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.797504719\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 49T \) |
| good | 3 | \( 1 + 16.3T + 243T^{2} \) |
| 11 | \( 1 - 728.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 305.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 490.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.10e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.19e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 3.73e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.67e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.57e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 2.23e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.40e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.15e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.73e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.66e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.80e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.89e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 9.81e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 7.54e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.88e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.54e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 8.00e3T + 5.58e9T^{2} \) |
| 97 | \( 1 - 3.20e4T + 8.58e9T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.669720048018853581456996378074, −8.897424909028997020196728595616, −7.982032636028197555823457464982, −6.58908121125031875380567048536, −6.36541552833779991818548424346, −5.18655113159686412013562983763, −4.35342779876872697063923664773, −3.22222657490113502271019941147, −1.56628012575627067162839479238, −0.72177477144317170830786974482,
0.72177477144317170830786974482, 1.56628012575627067162839479238, 3.22222657490113502271019941147, 4.35342779876872697063923664773, 5.18655113159686412013562983763, 6.36541552833779991818548424346, 6.58908121125031875380567048536, 7.982032636028197555823457464982, 8.897424909028997020196728595616, 9.669720048018853581456996378074
