L(s) = 1 | + 2.41·3-s + 2.82·9-s + 1.82·11-s + 6.41·13-s + 3.58·17-s + 7.65·19-s − 3.41·23-s − 0.414·27-s − 4.65·29-s − 7.41·31-s + 4.41·33-s + 0.585·37-s + 15.4·39-s − 3.41·41-s − 0.343·43-s + 10.8·47-s + 8.65·51-s + 12.2·53-s + 18.4·57-s − 0.585·59-s − 10.8·61-s − 3.07·67-s − 8.24·69-s − 10.4·71-s + 10.8·73-s − 15.1·79-s − 9.48·81-s + ⋯ |
L(s) = 1 | + 1.39·3-s + 0.942·9-s + 0.551·11-s + 1.77·13-s + 0.869·17-s + 1.75·19-s − 0.711·23-s − 0.0797·27-s − 0.864·29-s − 1.33·31-s + 0.768·33-s + 0.0963·37-s + 2.47·39-s − 0.533·41-s − 0.0523·43-s + 1.58·47-s + 1.21·51-s + 1.68·53-s + 2.44·57-s − 0.0762·59-s − 1.38·61-s − 0.375·67-s − 0.992·69-s − 1.24·71-s + 1.26·73-s − 1.70·79-s − 1.05·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.963074439\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.963074439\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - 2.41T + 3T^{2} \) |
| 11 | \( 1 - 1.82T + 11T^{2} \) |
| 13 | \( 1 - 6.41T + 13T^{2} \) |
| 17 | \( 1 - 3.58T + 17T^{2} \) |
| 19 | \( 1 - 7.65T + 19T^{2} \) |
| 23 | \( 1 + 3.41T + 23T^{2} \) |
| 29 | \( 1 + 4.65T + 29T^{2} \) |
| 31 | \( 1 + 7.41T + 31T^{2} \) |
| 37 | \( 1 - 0.585T + 37T^{2} \) |
| 41 | \( 1 + 3.41T + 41T^{2} \) |
| 43 | \( 1 + 0.343T + 43T^{2} \) |
| 47 | \( 1 - 10.8T + 47T^{2} \) |
| 53 | \( 1 - 12.2T + 53T^{2} \) |
| 59 | \( 1 + 0.585T + 59T^{2} \) |
| 61 | \( 1 + 10.8T + 61T^{2} \) |
| 67 | \( 1 + 3.07T + 67T^{2} \) |
| 71 | \( 1 + 10.4T + 71T^{2} \) |
| 73 | \( 1 - 10.8T + 73T^{2} \) |
| 79 | \( 1 + 15.1T + 79T^{2} \) |
| 83 | \( 1 + 8T + 83T^{2} \) |
| 89 | \( 1 - 16.9T + 89T^{2} \) |
| 97 | \( 1 + 9.72T + 97T^{2} \) |
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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
−8.361251205470429203853807829585, −7.56714521937640808791754525658, −7.18736704822174484613294152477, −5.92971230632405879527100337486, −5.55664483963845982611713316925, −4.15304975336719562120775089236, −3.56787507644199132378158165433, −3.11357139442858685877803954465, −1.90588883351041407079731985295, −1.13248615436590951685514244052,
1.13248615436590951685514244052, 1.90588883351041407079731985295, 3.11357139442858685877803954465, 3.56787507644199132378158165433, 4.15304975336719562120775089236, 5.55664483963845982611713316925, 5.92971230632405879527100337486, 7.18736704822174484613294152477, 7.56714521937640808791754525658, 8.361251205470429203853807829585