| L(s) = 1 | + 7-s + 4·11-s + 2·17-s + 6·19-s + 6·23-s + 2·31-s + 2·37-s − 2·41-s + 4·43-s + 8·47-s + 49-s − 10·53-s + 4·59-s + 2·61-s + 12·67-s − 8·71-s − 8·73-s + 4·77-s − 8·79-s + 4·83-s − 10·89-s + 4·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.377·7-s + 1.20·11-s + 0.485·17-s + 1.37·19-s + 1.25·23-s + 0.359·31-s + 0.328·37-s − 0.312·41-s + 0.609·43-s + 1.16·47-s + 1/7·49-s − 1.37·53-s + 0.520·59-s + 0.256·61-s + 1.46·67-s − 0.949·71-s − 0.936·73-s + 0.455·77-s − 0.900·79-s + 0.439·83-s − 1.05·89-s + 0.406·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.960269875\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.960269875\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−13.81135889025001, −13.33274925963928, −12.71826227345233, −12.18187499264739, −11.83862485019531, −11.26256156609594, −10.99688552017494, −10.26774361182812, −9.648660312957851, −9.397429556290067, −8.792890860884676, −8.324773649978357, −7.669771292888041, −7.178018158167234, −6.798588301538715, −6.093528965033474, −5.539572105221935, −5.086988677595991, −4.377559498132411, −3.937227406358306, −3.156543484938042, −2.810184633156306, −1.808844317501333, −1.202089971048016, −0.7124624447417337,
0.7124624447417337, 1.202089971048016, 1.808844317501333, 2.810184633156306, 3.156543484938042, 3.937227406358306, 4.377559498132411, 5.086988677595991, 5.539572105221935, 6.093528965033474, 6.798588301538715, 7.178018158167234, 7.669771292888041, 8.324773649978357, 8.792890860884676, 9.397429556290067, 9.648660312957851, 10.26774361182812, 10.99688552017494, 11.26256156609594, 11.83862485019531, 12.18187499264739, 12.71826227345233, 13.33274925963928, 13.81135889025001
