| L(s) = 1 | − 1.41·3-s − 5-s + 1.00·9-s − 1.41·13-s + 1.41·15-s + 19-s + 25-s + 1.41·37-s + 2.00·39-s − 1.00·45-s + 49-s + 1.41·53-s − 1.41·57-s + 1.41·65-s + 1.41·67-s − 1.41·75-s − 0.999·81-s − 95-s + 1.41·97-s + 1.41·103-s + 1.41·107-s − 2.00·111-s − 1.41·113-s − 1.41·117-s + ⋯ |
| L(s) = 1 | − 1.41·3-s − 5-s + 1.00·9-s − 1.41·13-s + 1.41·15-s + 19-s + 25-s + 1.41·37-s + 2.00·39-s − 1.00·45-s + 49-s + 1.41·53-s − 1.41·57-s + 1.41·65-s + 1.41·67-s − 1.41·75-s − 0.999·81-s − 95-s + 1.41·97-s + 1.41·103-s + 1.41·107-s − 2.00·111-s − 1.41·113-s − 1.41·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4822120654\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4822120654\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 1.41T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + 1.41T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - 1.41T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - 1.41T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 - 1.41T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - 1.41T + T^{2} \) |
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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
−9.881810391397916413009704096473, −8.909548534002982525343825109689, −7.75459383781529015979927766789, −7.26643416242760660844775979909, −6.42014215907750555193389890945, −5.40161300898695646180400397751, −4.84026935961743966237200129829, −3.93035808523073717032481274014, −2.64440395423594903169420657159, −0.76980421142416509575931543624,
0.76980421142416509575931543624, 2.64440395423594903169420657159, 3.93035808523073717032481274014, 4.84026935961743966237200129829, 5.40161300898695646180400397751, 6.42014215907750555193389890945, 7.26643416242760660844775979909, 7.75459383781529015979927766789, 8.909548534002982525343825109689, 9.881810391397916413009704096473
