L(s) = 1 | − 1.73·5-s − 7-s − 1.73·11-s + 1.73·17-s + 19-s + 1.99·25-s + 1.73·35-s + 43-s + 1.73·47-s + 2.99·55-s + 61-s − 73-s + 1.73·77-s − 2.99·85-s − 1.73·95-s − 1.73·119-s + ⋯ |
L(s) = 1 | − 1.73·5-s − 7-s − 1.73·11-s + 1.73·17-s + 19-s + 1.99·25-s + 1.73·35-s + 43-s + 1.73·47-s + 2.99·55-s + 61-s − 73-s + 1.73·77-s − 2.99·85-s − 1.73·95-s − 1.73·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6351601021\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6351601021\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 5 | \( 1 + 1.73T + T^{2} \) |
| 7 | \( 1 + T + T^{2} \) |
| 11 | \( 1 + 1.73T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - 1.73T + T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T + T^{2} \) |
| 47 | \( 1 - 1.73T + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 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
−8.900703449638181422581573172415, −7.961290920602328844846940300758, −7.63189394962964192311821693812, −7.04953657515592276335939624692, −5.79248924258483129659513496169, −5.16862967248409659090433585657, −4.10286384222054149602629525690, −3.29625699884825631953904319194, −2.78442535581043261675241404688, −0.70888517576267816430448333051,
0.70888517576267816430448333051, 2.78442535581043261675241404688, 3.29625699884825631953904319194, 4.10286384222054149602629525690, 5.16862967248409659090433585657, 5.79248924258483129659513496169, 7.04953657515592276335939624692, 7.63189394962964192311821693812, 7.961290920602328844846940300758, 8.900703449638181422581573172415