L(s) = 1 | + 2-s + 4-s − 5-s + 4·7-s + 8-s − 10-s + 6·13-s + 4·14-s + 16-s + 6·17-s + 6·19-s − 20-s − 8·23-s + 25-s + 6·26-s + 4·28-s − 6·29-s − 6·31-s + 32-s + 6·34-s − 4·35-s − 37-s + 6·38-s − 40-s − 10·43-s − 8·46-s + 8·47-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 1.51·7-s + 0.353·8-s − 0.316·10-s + 1.66·13-s + 1.06·14-s + 1/4·16-s + 1.45·17-s + 1.37·19-s − 0.223·20-s − 1.66·23-s + 1/5·25-s + 1.17·26-s + 0.755·28-s − 1.11·29-s − 1.07·31-s + 0.176·32-s + 1.02·34-s − 0.676·35-s − 0.164·37-s + 0.973·38-s − 0.158·40-s − 1.52·43-s − 1.17·46-s + 1.16·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.774332670\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.774332670\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 37 | \( 1 + T \) |
good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 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
−8.323228010474586230883593197400, −7.83916509293533494100581009678, −7.33053498998754819979853088298, −6.09858610918177968812596673686, −5.52349373963261666674358411316, −4.86907554365621367135840527494, −3.74744714998078559225470299488, −3.49528592717477161546933552500, −1.93143601746045232341937211043, −1.18570928018448164103325681997,
1.18570928018448164103325681997, 1.93143601746045232341937211043, 3.49528592717477161546933552500, 3.74744714998078559225470299488, 4.86907554365621367135840527494, 5.52349373963261666674358411316, 6.09858610918177968812596673686, 7.33053498998754819979853088298, 7.83916509293533494100581009678, 8.323228010474586230883593197400