L(s) = 1 | + 3-s − 3.21·5-s − 7-s + 9-s − 3.79·11-s − 6.57·13-s − 3.21·15-s + 0.454·17-s − 1.54·19-s − 21-s − 23-s + 5.33·25-s + 27-s − 3.36·29-s − 4.88·31-s − 3.79·33-s + 3.21·35-s + 4.88·37-s − 6.57·39-s − 0.883·41-s − 5.00·43-s − 3.21·45-s − 4.42·47-s + 49-s + 0.454·51-s + 0.123·53-s + 12.1·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 1.43·5-s − 0.377·7-s + 0.333·9-s − 1.14·11-s − 1.82·13-s − 0.829·15-s + 0.110·17-s − 0.354·19-s − 0.218·21-s − 0.208·23-s + 1.06·25-s + 0.192·27-s − 0.624·29-s − 0.877·31-s − 0.660·33-s + 0.543·35-s + 0.802·37-s − 1.05·39-s − 0.137·41-s − 0.763·43-s − 0.479·45-s − 0.645·47-s + 0.142·49-s + 0.0636·51-s + 0.0170·53-s + 1.64·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5202704672\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5202704672\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 5 | \( 1 + 3.21T + 5T^{2} \) |
| 11 | \( 1 + 3.79T + 11T^{2} \) |
| 13 | \( 1 + 6.57T + 13T^{2} \) |
| 17 | \( 1 - 0.454T + 17T^{2} \) |
| 19 | \( 1 + 1.54T + 19T^{2} \) |
| 29 | \( 1 + 3.36T + 29T^{2} \) |
| 31 | \( 1 + 4.88T + 31T^{2} \) |
| 37 | \( 1 - 4.88T + 37T^{2} \) |
| 41 | \( 1 + 0.883T + 41T^{2} \) |
| 43 | \( 1 + 5.00T + 43T^{2} \) |
| 47 | \( 1 + 4.42T + 47T^{2} \) |
| 53 | \( 1 - 0.123T + 53T^{2} \) |
| 59 | \( 1 - 5.87T + 59T^{2} \) |
| 61 | \( 1 + 8.33T + 61T^{2} \) |
| 67 | \( 1 + 1.87T + 67T^{2} \) |
| 71 | \( 1 - 13.2T + 71T^{2} \) |
| 73 | \( 1 + 5.13T + 73T^{2} \) |
| 79 | \( 1 + 11.1T + 79T^{2} \) |
| 83 | \( 1 + 1.79T + 83T^{2} \) |
| 89 | \( 1 - 0.150T + 89T^{2} \) |
| 97 | \( 1 - 9.61T + 97T^{2} \) |
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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
−7.82713227222982077365012894613, −7.36475129666248187744644476244, −6.82096482588298673040405124894, −5.64957256346011092664150617399, −4.87572433866845635562140879823, −4.28674609710486782548160537392, −3.44637116159750082615850509267, −2.81030178753131464128305461195, −2.01389359606571696671366565934, −0.32420786596393055999396332525,
0.32420786596393055999396332525, 2.01389359606571696671366565934, 2.81030178753131464128305461195, 3.44637116159750082615850509267, 4.28674609710486782548160537392, 4.87572433866845635562140879823, 5.64957256346011092664150617399, 6.82096482588298673040405124894, 7.36475129666248187744644476244, 7.82713227222982077365012894613