| L(s) = 1 | − 2-s − 3.10·3-s + 4-s − 3.25·5-s + 3.10·6-s + 4.58·7-s − 8-s + 6.65·9-s + 3.25·10-s + 0.524·11-s − 3.10·12-s + 0.0238·13-s − 4.58·14-s + 10.1·15-s + 16-s − 6.08·17-s − 6.65·18-s − 19-s − 3.25·20-s − 14.2·21-s − 0.524·22-s − 3.03·23-s + 3.10·24-s + 5.61·25-s − 0.0238·26-s − 11.3·27-s + 4.58·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.79·3-s + 0.5·4-s − 1.45·5-s + 1.26·6-s + 1.73·7-s − 0.353·8-s + 2.21·9-s + 1.03·10-s + 0.158·11-s − 0.896·12-s + 0.00661·13-s − 1.22·14-s + 2.61·15-s + 0.250·16-s − 1.47·17-s − 1.56·18-s − 0.229·19-s − 0.728·20-s − 3.11·21-s − 0.111·22-s − 0.631·23-s + 0.634·24-s + 1.12·25-s − 0.00467·26-s − 2.18·27-s + 0.866·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4815358532\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4815358532\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 - T \) |
| good | 3 | \( 1 + 3.10T + 3T^{2} \) |
| 5 | \( 1 + 3.25T + 5T^{2} \) |
| 7 | \( 1 - 4.58T + 7T^{2} \) |
| 11 | \( 1 - 0.524T + 11T^{2} \) |
| 13 | \( 1 - 0.0238T + 13T^{2} \) |
| 17 | \( 1 + 6.08T + 17T^{2} \) |
| 23 | \( 1 + 3.03T + 23T^{2} \) |
| 29 | \( 1 - 2.34T + 29T^{2} \) |
| 31 | \( 1 - 4.38T + 31T^{2} \) |
| 37 | \( 1 - 2.80T + 37T^{2} \) |
| 41 | \( 1 + 0.960T + 41T^{2} \) |
| 43 | \( 1 + 1.64T + 43T^{2} \) |
| 47 | \( 1 - 9.36T + 47T^{2} \) |
| 53 | \( 1 - 9.05T + 53T^{2} \) |
| 59 | \( 1 + 8.13T + 59T^{2} \) |
| 61 | \( 1 + 6.52T + 61T^{2} \) |
| 67 | \( 1 - 1.80T + 67T^{2} \) |
| 71 | \( 1 + 1.92T + 71T^{2} \) |
| 73 | \( 1 - 2.74T + 73T^{2} \) |
| 79 | \( 1 + 1.95T + 79T^{2} \) |
| 83 | \( 1 + 0.196T + 83T^{2} \) |
| 89 | \( 1 - 0.0816T + 89T^{2} \) |
| 97 | \( 1 - 11.6T + 97T^{2} \) |
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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
−7.70383923190008199590888651398, −7.26812260736224427799593944709, −6.52599242639053927942765303647, −5.85003317423800485645250891986, −4.87980593534238192511137121934, −4.49879387502391384528521266883, −3.92798889843422429309239280577, −2.31295904383629814208133835449, −1.31307269013824755007189343922, −0.47258129321492006598710618940,
0.47258129321492006598710618940, 1.31307269013824755007189343922, 2.31295904383629814208133835449, 3.92798889843422429309239280577, 4.49879387502391384528521266883, 4.87980593534238192511137121934, 5.85003317423800485645250891986, 6.52599242639053927942765303647, 7.26812260736224427799593944709, 7.70383923190008199590888651398
