L(s) = 1 | − i·2-s + (1.60 − 1.60i)3-s − 4-s + (0.707 − 0.707i)5-s + (−1.60 − 1.60i)6-s + (2.26 + 2.26i)7-s + i·8-s − 2.14i·9-s + (−0.707 − 0.707i)10-s + (−1.82 − 1.82i)11-s + (−1.60 + 1.60i)12-s − 5.68·13-s + (2.26 − 2.26i)14-s − 2.26i·15-s + 16-s + (0.604 + 4.07i)17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + (0.926 − 0.926i)3-s − 0.5·4-s + (0.316 − 0.316i)5-s + (−0.654 − 0.654i)6-s + (0.857 + 0.857i)7-s + 0.353i·8-s − 0.715i·9-s + (−0.223 − 0.223i)10-s + (−0.551 − 0.551i)11-s + (−0.463 + 0.463i)12-s − 1.57·13-s + (0.606 − 0.606i)14-s − 0.585i·15-s + 0.250·16-s + (0.146 + 0.989i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0245 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0245 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.06594 - 1.04010i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.06594 - 1.04010i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 5 | \( 1 + (-0.707 + 0.707i)T \) |
| 17 | \( 1 + (-0.604 - 4.07i)T \) |
good | 3 | \( 1 + (-1.60 + 1.60i)T - 3iT^{2} \) |
| 7 | \( 1 + (-2.26 - 2.26i)T + 7iT^{2} \) |
| 11 | \( 1 + (1.82 + 1.82i)T + 11iT^{2} \) |
| 13 | \( 1 + 5.68T + 13T^{2} \) |
| 19 | \( 1 - 4.35iT - 19T^{2} \) |
| 23 | \( 1 + (3.41 + 3.41i)T + 23iT^{2} \) |
| 29 | \( 1 + (-6.01 + 6.01i)T - 29iT^{2} \) |
| 31 | \( 1 + (1.16 - 1.16i)T - 31iT^{2} \) |
| 37 | \( 1 + (-3.09 + 3.09i)T - 37iT^{2} \) |
| 41 | \( 1 + (-5.53 - 5.53i)T + 41iT^{2} \) |
| 43 | \( 1 - 7.20iT - 43T^{2} \) |
| 47 | \( 1 + 4.89T + 47T^{2} \) |
| 53 | \( 1 + 1.14iT - 53T^{2} \) |
| 59 | \( 1 + 8.01iT - 59T^{2} \) |
| 61 | \( 1 + (4.51 + 4.51i)T + 61iT^{2} \) |
| 67 | \( 1 + 14.0T + 67T^{2} \) |
| 71 | \( 1 + (4.37 - 4.37i)T - 71iT^{2} \) |
| 73 | \( 1 + (-3.62 + 3.62i)T - 73iT^{2} \) |
| 79 | \( 1 + (1.29 + 1.29i)T + 79iT^{2} \) |
| 83 | \( 1 + 12.1iT - 83T^{2} \) |
| 89 | \( 1 - 16.2T + 89T^{2} \) |
| 97 | \( 1 + (12.5 - 12.5i)T - 97iT^{2} \) |
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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
−12.50807741489440927003174769098, −11.90664729692119830682302996141, −10.50464406893502965531443003600, −9.429937576737181083681983957145, −8.155535018386360743518581329524, −7.988443442900586450041162839848, −6.02596376209032578889050019626, −4.70079552104114368227734342227, −2.73658967997311696735169478097, −1.84068066408245972362613001080,
2.73645720009660852490064048077, 4.38932068598447730550770273819, 5.11339095268970229176873326281, 7.10962305155156079788852563782, 7.72146793648927585253552344953, 9.034840822914222927150802003966, 9.859326569828399363161885892289, 10.59604013823463688419843174026, 12.07885349706241551007891917026, 13.56871179282570640169069509804