L(s) = 1 | − 6i·5-s − 5i·7-s + 6·11-s − 5.19i·13-s − 31.1·17-s − 25.9·19-s − 31.1i·23-s − 11·25-s + 36i·29-s + 26i·31-s − 30·35-s + 25.9i·37-s − 20.7·43-s − 31.1i·47-s + 24·49-s + ⋯ |
L(s) = 1 | − 1.20i·5-s − 0.714i·7-s + 0.545·11-s − 0.399i·13-s − 1.83·17-s − 1.36·19-s − 1.35i·23-s − 0.440·25-s + 1.24i·29-s + 0.838i·31-s − 0.857·35-s + 0.702i·37-s − 0.483·43-s − 0.663i·47-s + 0.489·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3316860112\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3316860112\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 6iT - 25T^{2} \) |
| 7 | \( 1 + 5iT - 49T^{2} \) |
| 11 | \( 1 - 6T + 121T^{2} \) |
| 13 | \( 1 + 5.19iT - 169T^{2} \) |
| 17 | \( 1 + 31.1T + 289T^{2} \) |
| 19 | \( 1 + 25.9T + 361T^{2} \) |
| 23 | \( 1 + 31.1iT - 529T^{2} \) |
| 29 | \( 1 - 36iT - 841T^{2} \) |
| 31 | \( 1 - 26iT - 961T^{2} \) |
| 37 | \( 1 - 25.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 1.68e3T^{2} \) |
| 43 | \( 1 + 20.7T + 1.84e3T^{2} \) |
| 47 | \( 1 + 31.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 60iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 18T + 3.48e3T^{2} \) |
| 61 | \( 1 - 77.9iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 77.9T + 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 25T + 5.32e3T^{2} \) |
| 79 | \( 1 + 31iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 120T + 6.88e3T^{2} \) |
| 89 | \( 1 - 155.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 85T + 9.40e3T^{2} \) |
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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
−8.819712361028532635062005568468, −8.085695689094389407064806810480, −6.81753780882897922039367727588, −6.47563444441201173476082539906, −5.08644502037007069735321302136, −4.54614632437950421682803326819, −3.78520773864090544713815829100, −2.33064506367500722962377051374, −1.18305429127662150496365510566, −0.088058742603767786464212094786,
1.96955423347989828952921317692, 2.57351572884541179112075103749, 3.79032042766077811432701743923, 4.53621477639588217910757467323, 5.89822553355396595882968119072, 6.41375517829453595428098809524, 7.09793040932488428381268462789, 8.035042923831095306590329559599, 9.003822528309394924748605836456, 9.455533611312397244716535233474