L(s) = 1 | − 3·5-s + 7-s + 3·11-s + 13-s − 19-s − 23-s + 4·25-s + 6·29-s − 4·31-s − 3·35-s − 10·37-s − 10·43-s + 6·47-s + 49-s + 12·53-s − 9·55-s + 6·59-s − 4·61-s − 3·65-s + 5·67-s + 2·73-s + 3·77-s + 14·79-s − 15·83-s + 6·89-s + 91-s + 3·95-s + ⋯ |
L(s) = 1 | − 1.34·5-s + 0.377·7-s + 0.904·11-s + 0.277·13-s − 0.229·19-s − 0.208·23-s + 4/5·25-s + 1.11·29-s − 0.718·31-s − 0.507·35-s − 1.64·37-s − 1.52·43-s + 0.875·47-s + 1/7·49-s + 1.64·53-s − 1.21·55-s + 0.781·59-s − 0.512·61-s − 0.372·65-s + 0.610·67-s + 0.234·73-s + 0.341·77-s + 1.57·79-s − 1.64·83-s + 0.635·89-s + 0.104·91-s + 0.307·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75348 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75348 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−14.35748081968942, −13.81639886961338, −13.44522539925128, −12.59472010423755, −12.15746610382605, −11.92405554999469, −11.32523514646118, −10.99167550454004, −10.29976961414159, −9.911629914989862, −9.001471004450647, −8.665713880847518, −8.287822004016159, −7.685106915002936, −7.065273417413511, −6.772229778314719, −6.078471781213578, −5.300482785654255, −4.839720930627466, −4.040868382805125, −3.830878206621972, −3.239839799233554, −2.364116221955884, −1.587078341121003, −0.8570444555565090, 0,
0.8570444555565090, 1.587078341121003, 2.364116221955884, 3.239839799233554, 3.830878206621972, 4.040868382805125, 4.839720930627466, 5.300482785654255, 6.078471781213578, 6.772229778314719, 7.065273417413511, 7.685106915002936, 8.287822004016159, 8.665713880847518, 9.001471004450647, 9.911629914989862, 10.29976961414159, 10.99167550454004, 11.32523514646118, 11.92405554999469, 12.15746610382605, 12.59472010423755, 13.44522539925128, 13.81639886961338, 14.35748081968942