| L(s) = 1 | − 0.249·3-s − 3.41·5-s + 3.14·7-s − 2.93·9-s + 4.41·11-s + 5.12·13-s + 0.853·15-s + 6.01·17-s + 5.70·19-s − 0.785·21-s − 3.81·23-s + 6.69·25-s + 1.48·27-s − 1.63·29-s − 3.23·31-s − 1.10·33-s − 10.7·35-s + 4.58·37-s − 1.27·39-s + 8.55·41-s + 7.40·43-s + 10.0·45-s − 0.286·47-s + 2.89·49-s − 1.50·51-s − 2.90·53-s − 15.1·55-s + ⋯ |
| L(s) = 1 | − 0.144·3-s − 1.52·5-s + 1.18·7-s − 0.979·9-s + 1.33·11-s + 1.42·13-s + 0.220·15-s + 1.45·17-s + 1.30·19-s − 0.171·21-s − 0.794·23-s + 1.33·25-s + 0.285·27-s − 0.302·29-s − 0.580·31-s − 0.191·33-s − 1.81·35-s + 0.753·37-s − 0.204·39-s + 1.33·41-s + 1.13·43-s + 1.49·45-s − 0.0417·47-s + 0.414·49-s − 0.210·51-s − 0.399·53-s − 2.03·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.051553884\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.051553884\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 503 | \( 1 + T \) |
| good | 3 | \( 1 + 0.249T + 3T^{2} \) |
| 5 | \( 1 + 3.41T + 5T^{2} \) |
| 7 | \( 1 - 3.14T + 7T^{2} \) |
| 11 | \( 1 - 4.41T + 11T^{2} \) |
| 13 | \( 1 - 5.12T + 13T^{2} \) |
| 17 | \( 1 - 6.01T + 17T^{2} \) |
| 19 | \( 1 - 5.70T + 19T^{2} \) |
| 23 | \( 1 + 3.81T + 23T^{2} \) |
| 29 | \( 1 + 1.63T + 29T^{2} \) |
| 31 | \( 1 + 3.23T + 31T^{2} \) |
| 37 | \( 1 - 4.58T + 37T^{2} \) |
| 41 | \( 1 - 8.55T + 41T^{2} \) |
| 43 | \( 1 - 7.40T + 43T^{2} \) |
| 47 | \( 1 + 0.286T + 47T^{2} \) |
| 53 | \( 1 + 2.90T + 53T^{2} \) |
| 59 | \( 1 + 5.10T + 59T^{2} \) |
| 61 | \( 1 - 1.08T + 61T^{2} \) |
| 67 | \( 1 + 2.78T + 67T^{2} \) |
| 71 | \( 1 - 1.27T + 71T^{2} \) |
| 73 | \( 1 - 12.0T + 73T^{2} \) |
| 79 | \( 1 - 14.0T + 79T^{2} \) |
| 83 | \( 1 + 10.3T + 83T^{2} \) |
| 89 | \( 1 + 1.69T + 89T^{2} \) |
| 97 | \( 1 + 2.26T + 97T^{2} \) |
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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
−7.84690106611058651461478385215, −7.47242915547366672107488212886, −6.36628798169113284154639563244, −5.73786521244977556227670876696, −5.04735982413053134165065124349, −3.94323628796031292038497899033, −3.82509982750303332790114799565, −2.90198962616023894347544117987, −1.43267679999232756076807277212, −0.816057779540868095847442961526,
0.816057779540868095847442961526, 1.43267679999232756076807277212, 2.90198962616023894347544117987, 3.82509982750303332790114799565, 3.94323628796031292038497899033, 5.04735982413053134165065124349, 5.73786521244977556227670876696, 6.36628798169113284154639563244, 7.47242915547366672107488212886, 7.84690106611058651461478385215
