| L(s) = 1 | + 2-s + 4-s − 4·7-s + 8-s − 4·13-s − 4·14-s + 16-s − 17-s − 4·19-s − 5·25-s − 4·26-s − 4·28-s − 4·31-s + 32-s − 34-s − 7·37-s − 4·38-s + 3·41-s − 10·43-s − 6·47-s + 9·49-s − 5·50-s − 4·52-s − 3·53-s − 4·56-s + 9·59-s − 7·61-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.51·7-s + 0.353·8-s − 1.10·13-s − 1.06·14-s + 1/4·16-s − 0.242·17-s − 0.917·19-s − 25-s − 0.784·26-s − 0.755·28-s − 0.718·31-s + 0.176·32-s − 0.171·34-s − 1.15·37-s − 0.648·38-s + 0.468·41-s − 1.52·43-s − 0.875·47-s + 9/7·49-s − 0.707·50-s − 0.554·52-s − 0.412·53-s − 0.534·56-s + 1.17·59-s − 0.896·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7146568421\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7146568421\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−15.01891246356088, −14.35078040822251, −13.74252952302574, −13.19191079582258, −12.90565167498211, −12.34502014027061, −11.92048522958218, −11.33370737431541, −10.59291177879371, −10.07949568553085, −9.737205039615858, −9.067447191207974, −8.485667363334769, −7.651977091204074, −7.163996481823982, −6.571746237791966, −6.201293741866699, −5.506376479903316, −4.892541215084221, −4.219594783634039, −3.550476881135829, −3.086862358907915, −2.315007997736511, −1.718506485219893, −0.2548201305021798,
0.2548201305021798, 1.718506485219893, 2.315007997736511, 3.086862358907915, 3.550476881135829, 4.219594783634039, 4.892541215084221, 5.506376479903316, 6.201293741866699, 6.571746237791966, 7.163996481823982, 7.651977091204074, 8.485667363334769, 9.067447191207974, 9.737205039615858, 10.07949568553085, 10.59291177879371, 11.33370737431541, 11.92048522958218, 12.34502014027061, 12.90565167498211, 13.19191079582258, 13.74252952302574, 14.35078040822251, 15.01891246356088
