L(s) = 1 | − 2-s + 4-s + 5-s − 8-s − 10-s − 13-s + 16-s + 17-s + 20-s + 26-s + 29-s − 32-s − 34-s − 37-s − 40-s − 2·41-s + 49-s − 52-s − 2·53-s − 58-s − 61-s + 64-s − 65-s + 68-s − 73-s + 74-s + 80-s + ⋯ |
L(s) = 1 | − 2-s + 4-s + 5-s − 8-s − 10-s − 13-s + 16-s + 17-s + 20-s + 26-s + 29-s − 32-s − 34-s − 37-s − 40-s − 2·41-s + 49-s − 52-s − 2·53-s − 58-s − 61-s + 64-s − 65-s + 68-s − 73-s + 74-s + 80-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6023960610\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6023960610\) |
\(L(1)\) |
|
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 | \( 1 \) |
good | 5 | \( 1 - T + T^{2} \) |
| 7 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( 1 + T + T^{2} \) |
| 17 | \( 1 - T + T^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( 1 - T + T^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( ( 1 + T )^{2} \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 + T )^{2} \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( 1 - T + T^{2} \) |
| 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
−11.81710346471981409701953433839, −10.48981173802154940702996924690, −9.988446417078204397418780787230, −9.203979896353471130975111024289, −8.152183121051760303376962197843, −7.15124242104491554465471503376, −6.16229398107777236119729349046, −5.12295067737109190780447376161, −3.06897421321693433842888421737, −1.76200125015610031025475652937,
1.76200125015610031025475652937, 3.06897421321693433842888421737, 5.12295067737109190780447376161, 6.16229398107777236119729349046, 7.15124242104491554465471503376, 8.152183121051760303376962197843, 9.203979896353471130975111024289, 9.988446417078204397418780787230, 10.48981173802154940702996924690, 11.81710346471981409701953433839