| L(s) = 1 | + 2-s + 4-s + 4·7-s + 8-s + 2·11-s + 6·13-s + 4·14-s + 16-s − 17-s + 4·19-s + 2·22-s + 5·23-s + 6·26-s + 4·28-s + 10·31-s + 32-s − 34-s − 9·37-s + 4·38-s − 11·41-s − 10·43-s + 2·44-s + 5·46-s + 8·47-s + 9·49-s + 6·52-s − 11·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.51·7-s + 0.353·8-s + 0.603·11-s + 1.66·13-s + 1.06·14-s + 1/4·16-s − 0.242·17-s + 0.917·19-s + 0.426·22-s + 1.04·23-s + 1.17·26-s + 0.755·28-s + 1.79·31-s + 0.176·32-s − 0.171·34-s − 1.47·37-s + 0.648·38-s − 1.71·41-s − 1.52·43-s + 0.301·44-s + 0.737·46-s + 1.16·47-s + 9/7·49-s + 0.832·52-s − 1.51·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.058224005\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.058224005\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 + 11 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 - 15 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + 11 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 5 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−7.985537860823507319364059454113, −6.88933726630694787909884555467, −6.61812811922214293051626196442, −5.51941040798251481003769363914, −5.12330525859265529469069967000, −4.33977824292271141341339784349, −3.64247995417089730355005364511, −2.85998336048520689944133568239, −1.59470488278465744334536289089, −1.21420924893543569221885428019,
1.21420924893543569221885428019, 1.59470488278465744334536289089, 2.85998336048520689944133568239, 3.64247995417089730355005364511, 4.33977824292271141341339784349, 5.12330525859265529469069967000, 5.51941040798251481003769363914, 6.61812811922214293051626196442, 6.88933726630694787909884555467, 7.985537860823507319364059454113
