L(s) = 1 | + 2-s + 4-s − 7-s + 8-s + 11-s − 2·13-s − 14-s + 16-s + 4·17-s + 6·19-s + 22-s + 4·23-s − 5·25-s − 2·26-s − 28-s + 10·29-s + 6·31-s + 32-s + 4·34-s − 6·37-s + 6·38-s + 12·41-s − 8·43-s + 44-s + 4·46-s − 2·47-s + 49-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s + 0.301·11-s − 0.554·13-s − 0.267·14-s + 1/4·16-s + 0.970·17-s + 1.37·19-s + 0.213·22-s + 0.834·23-s − 25-s − 0.392·26-s − 0.188·28-s + 1.85·29-s + 1.07·31-s + 0.176·32-s + 0.685·34-s − 0.986·37-s + 0.973·38-s + 1.87·41-s − 1.21·43-s + 0.150·44-s + 0.589·46-s − 0.291·47-s + 1/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.685417362\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.685417362\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
good | 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−9.891429834212501423422030953706, −8.751364050919664408948964913092, −7.76644875192378882330336596739, −7.06243874841004507589844210062, −6.19912680921072839463289428224, −5.34458459387631487464034223256, −4.55153014528148769824162116427, −3.42266397038887650714886536973, −2.70873132071359508119536298434, −1.15718775335031232118843306899,
1.15718775335031232118843306899, 2.70873132071359508119536298434, 3.42266397038887650714886536973, 4.55153014528148769824162116427, 5.34458459387631487464034223256, 6.19912680921072839463289428224, 7.06243874841004507589844210062, 7.76644875192378882330336596739, 8.751364050919664408948964913092, 9.891429834212501423422030953706