| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 2·7-s + 8-s + 9-s − 10-s − 2·11-s − 12-s − 6·13-s + 2·14-s + 15-s + 16-s + 18-s + 2·19-s − 20-s − 2·21-s − 2·22-s + 8·23-s − 24-s + 25-s − 6·26-s − 27-s + 2·28-s + 30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.755·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.603·11-s − 0.288·12-s − 1.66·13-s + 0.534·14-s + 0.258·15-s + 1/4·16-s + 0.235·18-s + 0.458·19-s − 0.223·20-s − 0.436·21-s − 0.426·22-s + 1.66·23-s − 0.204·24-s + 1/5·25-s − 1.17·26-s − 0.192·27-s + 0.377·28-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.506348160\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.506348160\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 41 | \( 1 \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−14.56125297536976, −14.14996127821660, −13.34365163701681, −12.90414816066624, −12.51676061575289, −11.86581508842036, −11.56598727590348, −11.08219800652076, −10.50430524235882, −10.03648038623924, −9.399749179296453, −8.688575604065179, −8.022274289941548, −7.441922948194483, −7.137142227844314, −6.600018157022275, −5.660954806363413, −5.154969547570941, −4.932724612287778, −4.334174217414928, −3.581430280097967, −2.783410574904845, −2.322582162372467, −1.372965227653644, −0.5238116703256305,
0.5238116703256305, 1.372965227653644, 2.322582162372467, 2.783410574904845, 3.581430280097967, 4.334174217414928, 4.932724612287778, 5.154969547570941, 5.660954806363413, 6.600018157022275, 7.137142227844314, 7.441922948194483, 8.022274289941548, 8.688575604065179, 9.399749179296453, 10.03648038623924, 10.50430524235882, 11.08219800652076, 11.56598727590348, 11.86581508842036, 12.51676061575289, 12.90414816066624, 13.34365163701681, 14.14996127821660, 14.56125297536976
