L(s) = 1 | + 3-s + 5-s − 3.12·7-s + 9-s + 3.12·11-s + 2·13-s + 15-s − 1.12·17-s − 4·19-s − 3.12·21-s − 23-s + 25-s + 27-s + 2·29-s + 3.12·33-s − 3.12·35-s + 1.12·37-s + 2·39-s + 2·41-s + 45-s + 8·47-s + 2.75·49-s − 1.12·51-s + 12.2·53-s + 3.12·55-s − 4·57-s − 2.24·59-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s − 1.18·7-s + 0.333·9-s + 0.941·11-s + 0.554·13-s + 0.258·15-s − 0.272·17-s − 0.917·19-s − 0.681·21-s − 0.208·23-s + 0.200·25-s + 0.192·27-s + 0.371·29-s + 0.543·33-s − 0.527·35-s + 0.184·37-s + 0.320·39-s + 0.312·41-s + 0.149·45-s + 1.16·47-s + 0.393·49-s − 0.157·51-s + 1.68·53-s + 0.421·55-s − 0.529·57-s − 0.292·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.469285115\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.469285115\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 7 | \( 1 + 3.12T + 7T^{2} \) |
| 11 | \( 1 - 3.12T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 1.12T + 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 1.12T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 - 12.2T + 53T^{2} \) |
| 59 | \( 1 + 2.24T + 59T^{2} \) |
| 61 | \( 1 - 9.12T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 - 10.2T + 71T^{2} \) |
| 73 | \( 1 + 4.24T + 73T^{2} \) |
| 79 | \( 1 + 3.12T + 79T^{2} \) |
| 83 | \( 1 - 13.3T + 83T^{2} \) |
| 89 | \( 1 + 5.12T + 89T^{2} \) |
| 97 | \( 1 + 16.2T + 97T^{2} \) |
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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
−8.378773464612410435369465851807, −7.33877564468612088455253059729, −6.57718184349462358510254592393, −6.27736588391756662698826127512, −5.36681555767448325692362727495, −4.16502675631748651459014362026, −3.75780652074364951223220137591, −2.79026582685072621342139630451, −2.02040657616326832427191301053, −0.820781935671373985353206185677,
0.820781935671373985353206185677, 2.02040657616326832427191301053, 2.79026582685072621342139630451, 3.75780652074364951223220137591, 4.16502675631748651459014362026, 5.36681555767448325692362727495, 6.27736588391756662698826127512, 6.57718184349462358510254592393, 7.33877564468612088455253059729, 8.378773464612410435369465851807