L(s) = 1 | + 3-s + 0.694·5-s + 0.305·7-s + 9-s + 4.82·11-s − 13-s + 0.694·15-s − 7.51·17-s + 0.305·21-s + 0.694·23-s − 4.51·25-s + 27-s − 10.1·29-s + 1.82·31-s + 4.82·33-s + 0.212·35-s − 6.51·37-s − 39-s − 5.38·41-s + 3.69·43-s + 0.694·45-s − 6·47-s − 6.90·49-s − 7.51·51-s − 5.43·53-s + 3.34·55-s + 4.08·59-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.310·5-s + 0.115·7-s + 0.333·9-s + 1.45·11-s − 0.277·13-s + 0.179·15-s − 1.82·17-s + 0.0666·21-s + 0.144·23-s − 0.903·25-s + 0.192·27-s − 1.88·29-s + 0.327·31-s + 0.839·33-s + 0.0358·35-s − 1.07·37-s − 0.160·39-s − 0.841·41-s + 0.563·43-s + 0.103·45-s − 0.875·47-s − 0.986·49-s − 1.05·51-s − 0.746·53-s + 0.451·55-s + 0.531·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 19 | \( 1 \) |
good | 5 | \( 1 - 0.694T + 5T^{2} \) |
| 7 | \( 1 - 0.305T + 7T^{2} \) |
| 11 | \( 1 - 4.82T + 11T^{2} \) |
| 13 | \( 1 + T + 13T^{2} \) |
| 17 | \( 1 + 7.51T + 17T^{2} \) |
| 23 | \( 1 - 0.694T + 23T^{2} \) |
| 29 | \( 1 + 10.1T + 29T^{2} \) |
| 31 | \( 1 - 1.82T + 31T^{2} \) |
| 37 | \( 1 + 6.51T + 37T^{2} \) |
| 41 | \( 1 + 5.38T + 41T^{2} \) |
| 43 | \( 1 - 3.69T + 43T^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 + 5.43T + 53T^{2} \) |
| 59 | \( 1 - 4.08T + 59T^{2} \) |
| 61 | \( 1 - 0.389T + 61T^{2} \) |
| 67 | \( 1 - 7.82T + 67T^{2} \) |
| 71 | \( 1 + 10.9T + 71T^{2} \) |
| 73 | \( 1 + 4.38T + 73T^{2} \) |
| 79 | \( 1 + 14.4T + 79T^{2} \) |
| 83 | \( 1 - 0.739T + 83T^{2} \) |
| 89 | \( 1 - 0.822T + 89T^{2} \) |
| 97 | \( 1 + 10.9T + 97T^{2} \) |
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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
−7.32111366365350074581422153174, −6.79078545380025192134497466910, −6.20847736599672451544253035399, −5.33398173846403903220510429581, −4.42712243405257729728345888730, −3.92397261493310377912322884568, −3.10175522928482241903127756479, −2.00201159338040773759171775047, −1.60909202142343944445916908517, 0,
1.60909202142343944445916908517, 2.00201159338040773759171775047, 3.10175522928482241903127756479, 3.92397261493310377912322884568, 4.42712243405257729728345888730, 5.33398173846403903220510429581, 6.20847736599672451544253035399, 6.79078545380025192134497466910, 7.32111366365350074581422153174