L(s) = 1 | − 2-s − 3-s + 4-s − 3.21·5-s + 6-s − 1.40·7-s − 8-s + 9-s + 3.21·10-s − 2.53·11-s − 12-s + 13-s + 1.40·14-s + 3.21·15-s + 16-s + 7.57·17-s − 18-s + 2.48·19-s − 3.21·20-s + 1.40·21-s + 2.53·22-s + 7.10·23-s + 24-s + 5.33·25-s − 26-s − 27-s − 1.40·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.43·5-s + 0.408·6-s − 0.531·7-s − 0.353·8-s + 0.333·9-s + 1.01·10-s − 0.764·11-s − 0.288·12-s + 0.277·13-s + 0.375·14-s + 0.830·15-s + 0.250·16-s + 1.83·17-s − 0.235·18-s + 0.569·19-s − 0.718·20-s + 0.306·21-s + 0.540·22-s + 1.48·23-s + 0.204·24-s + 1.06·25-s − 0.196·26-s − 0.192·27-s − 0.265·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6402686369\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6402686369\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 + T \) |
good | 5 | \( 1 + 3.21T + 5T^{2} \) |
| 7 | \( 1 + 1.40T + 7T^{2} \) |
| 11 | \( 1 + 2.53T + 11T^{2} \) |
| 17 | \( 1 - 7.57T + 17T^{2} \) |
| 19 | \( 1 - 2.48T + 19T^{2} \) |
| 23 | \( 1 - 7.10T + 23T^{2} \) |
| 29 | \( 1 + 1.09T + 29T^{2} \) |
| 31 | \( 1 + 2.34T + 31T^{2} \) |
| 37 | \( 1 - 5.11T + 37T^{2} \) |
| 41 | \( 1 + 0.317T + 41T^{2} \) |
| 43 | \( 1 - 9.60T + 43T^{2} \) |
| 47 | \( 1 + 5.52T + 47T^{2} \) |
| 53 | \( 1 + 4.04T + 53T^{2} \) |
| 59 | \( 1 - 0.591T + 59T^{2} \) |
| 61 | \( 1 + 0.0491T + 61T^{2} \) |
| 67 | \( 1 + 11.1T + 67T^{2} \) |
| 71 | \( 1 + 16.7T + 71T^{2} \) |
| 73 | \( 1 - 5.34T + 73T^{2} \) |
| 79 | \( 1 - 16.3T + 79T^{2} \) |
| 83 | \( 1 - 3.14T + 83T^{2} \) |
| 89 | \( 1 + 4.41T + 89T^{2} \) |
| 97 | \( 1 + 9.59T + 97T^{2} \) |
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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
−7.70753682739136832049616097687, −7.42498899410928935122183102910, −6.64069146383162846363881915193, −5.75980472462594208864190586046, −5.16236954197431322887815755235, −4.23775812098959791751018217350, −3.31045039073143783451033331805, −2.92613584974200004010742514198, −1.31633656835623084207108412431, −0.50590575184155791352023525741,
0.50590575184155791352023525741, 1.31633656835623084207108412431, 2.92613584974200004010742514198, 3.31045039073143783451033331805, 4.23775812098959791751018217350, 5.16236954197431322887815755235, 5.75980472462594208864190586046, 6.64069146383162846363881915193, 7.42498899410928935122183102910, 7.70753682739136832049616097687