L(s) = 1 | + 1.41·5-s + 1.41·13-s − 1.41·17-s + 4·23-s − 2.99·25-s + 6·29-s + 5.65·31-s − 4·37-s − 7.07·41-s + 4·43-s + 11.3·47-s + 4·53-s − 5.65·59-s + 4.24·61-s + 2.00·65-s + 4·67-s + 12·71-s − 7.07·73-s + 8·79-s − 5.65·83-s − 2.00·85-s + 12.7·89-s + 4.24·97-s − 9.89·101-s − 11.3·103-s + 16·107-s − 4·109-s + ⋯ |
L(s) = 1 | + 0.632·5-s + 0.392·13-s − 0.342·17-s + 0.834·23-s − 0.599·25-s + 1.11·29-s + 1.01·31-s − 0.657·37-s − 1.10·41-s + 0.609·43-s + 1.65·47-s + 0.549·53-s − 0.736·59-s + 0.543·61-s + 0.248·65-s + 0.488·67-s + 1.42·71-s − 0.827·73-s + 0.900·79-s − 0.620·83-s − 0.216·85-s + 1.34·89-s + 0.430·97-s − 0.985·101-s − 1.11·103-s + 1.54·107-s − 0.383·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.203680966\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.203680966\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 1.41T + 5T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 1.41T + 13T^{2} \) |
| 17 | \( 1 + 1.41T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 5.65T + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 + 7.07T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 - 11.3T + 47T^{2} \) |
| 53 | \( 1 - 4T + 53T^{2} \) |
| 59 | \( 1 + 5.65T + 59T^{2} \) |
| 61 | \( 1 - 4.24T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + 7.07T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 5.65T + 83T^{2} \) |
| 89 | \( 1 - 12.7T + 89T^{2} \) |
| 97 | \( 1 - 4.24T + 97T^{2} \) |
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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
−8.657284465829937159154535871358, −7.898343611211250987278091707209, −6.95208894495956377479898243742, −6.37388625516611858227023080884, −5.58192272729343979893803934832, −4.82833326096884863195989945874, −3.92842572480601102662905257371, −2.92312634098921623400413298936, −2.04358096292737612482560710779, −0.898270004494159331583658136594,
0.898270004494159331583658136594, 2.04358096292737612482560710779, 2.92312634098921623400413298936, 3.92842572480601102662905257371, 4.82833326096884863195989945874, 5.58192272729343979893803934832, 6.37388625516611858227023080884, 6.95208894495956377479898243742, 7.898343611211250987278091707209, 8.657284465829937159154535871358