L(s) = 1 | + 0.749·2-s + 3.38·3-s − 1.43·4-s − 5-s + 2.54·6-s + 1.17·7-s − 2.57·8-s + 8.48·9-s − 0.749·10-s − 11-s − 4.87·12-s + 4.06·13-s + 0.881·14-s − 3.38·15-s + 0.944·16-s + 3.98·17-s + 6.35·18-s + 19-s + 1.43·20-s + 3.98·21-s − 0.749·22-s − 7.49·23-s − 8.73·24-s + 25-s + 3.04·26-s + 18.5·27-s − 1.69·28-s + ⋯ |
L(s) = 1 | + 0.530·2-s + 1.95·3-s − 0.719·4-s − 0.447·5-s + 1.03·6-s + 0.444·7-s − 0.911·8-s + 2.82·9-s − 0.237·10-s − 0.301·11-s − 1.40·12-s + 1.12·13-s + 0.235·14-s − 0.874·15-s + 0.236·16-s + 0.965·17-s + 1.49·18-s + 0.229·19-s + 0.321·20-s + 0.869·21-s − 0.159·22-s − 1.56·23-s − 1.78·24-s + 0.200·25-s + 0.597·26-s + 3.57·27-s − 0.319·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.438278781\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.438278781\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 - T \) |
good | 2 | \( 1 - 0.749T + 2T^{2} \) |
| 3 | \( 1 - 3.38T + 3T^{2} \) |
| 7 | \( 1 - 1.17T + 7T^{2} \) |
| 13 | \( 1 - 4.06T + 13T^{2} \) |
| 17 | \( 1 - 3.98T + 17T^{2} \) |
| 23 | \( 1 + 7.49T + 23T^{2} \) |
| 29 | \( 1 - 6.17T + 29T^{2} \) |
| 31 | \( 1 - 1.69T + 31T^{2} \) |
| 37 | \( 1 + 10.1T + 37T^{2} \) |
| 41 | \( 1 + 0.578T + 41T^{2} \) |
| 43 | \( 1 + 10.4T + 43T^{2} \) |
| 47 | \( 1 - 1.86T + 47T^{2} \) |
| 53 | \( 1 - 6.85T + 53T^{2} \) |
| 59 | \( 1 - 11.8T + 59T^{2} \) |
| 61 | \( 1 + 9.92T + 61T^{2} \) |
| 67 | \( 1 - 12.6T + 67T^{2} \) |
| 71 | \( 1 + 3.05T + 71T^{2} \) |
| 73 | \( 1 + 11.8T + 73T^{2} \) |
| 79 | \( 1 + 10.1T + 79T^{2} \) |
| 83 | \( 1 + 5.00T + 83T^{2} \) |
| 89 | \( 1 + 4.29T + 89T^{2} \) |
| 97 | \( 1 - 7.53T + 97T^{2} \) |
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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
−9.873764932849893294204070847549, −8.737563216431785599013204452761, −8.370834894917612183104853371922, −7.84791786960117590079759421559, −6.70701766699734287142065118535, −5.33790953620492671425820101059, −4.26417372744287124463364415599, −3.65823244148270551154538549990, −2.91668964940122381104502785335, −1.47593249080548411893989374972,
1.47593249080548411893989374972, 2.91668964940122381104502785335, 3.65823244148270551154538549990, 4.26417372744287124463364415599, 5.33790953620492671425820101059, 6.70701766699734287142065118535, 7.84791786960117590079759421559, 8.370834894917612183104853371922, 8.737563216431785599013204452761, 9.873764932849893294204070847549