L(s) = 1 | + i·3-s + 4.13i·7-s − 9-s − 3.76·11-s − 0.698i·13-s + 5.29i·17-s − 5.73·19-s − 4.13·21-s + 5.46i·23-s − i·27-s + 7.02·29-s − 10.0·31-s − 3.76i·33-s − 5.78i·37-s + 0.698·39-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + 1.56i·7-s − 0.333·9-s − 1.13·11-s − 0.193i·13-s + 1.28i·17-s − 1.31·19-s − 0.901·21-s + 1.13i·23-s − 0.192i·27-s + 1.30·29-s − 1.81·31-s − 0.655i·33-s − 0.951i·37-s + 0.111·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1783509871\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1783509871\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 4.13iT - 7T^{2} \) |
| 11 | \( 1 + 3.76T + 11T^{2} \) |
| 13 | \( 1 + 0.698iT - 13T^{2} \) |
| 17 | \( 1 - 5.29iT - 17T^{2} \) |
| 19 | \( 1 + 5.73T + 19T^{2} \) |
| 23 | \( 1 - 5.46iT - 23T^{2} \) |
| 29 | \( 1 - 7.02T + 29T^{2} \) |
| 31 | \( 1 + 10.0T + 31T^{2} \) |
| 37 | \( 1 + 5.78iT - 37T^{2} \) |
| 41 | \( 1 + 6.59T + 41T^{2} \) |
| 43 | \( 1 - 4.79iT - 43T^{2} \) |
| 47 | \( 1 + 9.67iT - 47T^{2} \) |
| 53 | \( 1 + 3.39iT - 53T^{2} \) |
| 59 | \( 1 + 0.745T + 59T^{2} \) |
| 61 | \( 1 - 11.8T + 61T^{2} \) |
| 67 | \( 1 - 4.79iT - 67T^{2} \) |
| 71 | \( 1 + 3.14T + 71T^{2} \) |
| 73 | \( 1 - 10.7iT - 73T^{2} \) |
| 79 | \( 1 - 9.12T + 79T^{2} \) |
| 83 | \( 1 - 8.44iT - 83T^{2} \) |
| 89 | \( 1 - 3.38T + 89T^{2} \) |
| 97 | \( 1 + 10.4iT - 97T^{2} \) |
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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
−8.505293446766005297126285349341, −7.990166559515750587421600689683, −6.96251686229939796548430370251, −6.10781708165899575833695197188, −5.47651933493435982188986004043, −5.15975091049317853368354449873, −4.07661794184364127445808570261, −3.34799829781530999196592720303, −2.42399567026112278098495948589, −1.85669098740876048099679042175,
0.04958391623476246245839031962, 0.847906169603278746986612842447, 2.03359599926005932947459838794, 2.83630875389823867985841653527, 3.72515190868231650548757500845, 4.65702460884492789899059173467, 5.04576362009080057668681365735, 6.23802764295865868685718689827, 6.76497888457785218630767210426, 7.36144641914939293349191982918