L(s) = 1 | + 1.41i·2-s − 1.00·4-s − 1.41i·7-s − 11-s − 1.41i·13-s + 2.00·14-s − 0.999·16-s − 1.41i·17-s − 1.41i·22-s + 2.00·26-s + 1.41i·28-s − 1.41i·32-s + 2.00·34-s − 1.41i·43-s + 1.00·44-s + ⋯ |
L(s) = 1 | + 1.41i·2-s − 1.00·4-s − 1.41i·7-s − 11-s − 1.41i·13-s + 2.00·14-s − 0.999·16-s − 1.41i·17-s − 1.41i·22-s + 2.00·26-s + 1.41i·28-s − 1.41i·32-s + 2.00·34-s − 1.41i·43-s + 1.00·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9106919591\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9106919591\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 2 | \( 1 - 1.41iT - T^{2} \) |
| 7 | \( 1 + 1.41iT - T^{2} \) |
| 13 | \( 1 + 1.41iT - T^{2} \) |
| 17 | \( 1 + 1.41iT - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + 1.41iT - T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 - 1.41iT - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + 1.41iT - T^{2} \) |
| 89 | \( 1 - 2T + T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−8.796382487019720156064712316084, −8.008697454278127680228133237409, −7.42725775038231072163251504412, −7.11162967738839176791672166010, −6.07882341212895919678951994991, −5.24750873227289428694305017670, −4.76344684937300545934072862640, −3.59593173238698278212631421531, −2.53491261925341392823887334934, −0.58782236177049019704470515032,
1.66272384420624847595547089639, 2.29292202697258242983783521447, 3.12705762268618250485070417587, 4.12775050035510841839462947434, 4.95013718549775742299814285568, 5.94928391167267172541956313069, 6.64864046221455927308715424955, 7.86787089806435487365283782211, 8.665890775692716984870400346682, 9.255446755635192883616413321123