| L(s) = 1 | + 2-s + 4-s − 5-s + 8-s − 10-s − 13-s + 16-s − 20-s + 6·23-s + 25-s − 26-s + 6·29-s + 2·31-s + 32-s − 2·37-s − 40-s + 2·41-s + 8·43-s + 6·46-s + 12·47-s − 7·49-s + 50-s − 52-s − 8·53-s + 6·58-s − 8·59-s − 10·61-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.353·8-s − 0.316·10-s − 0.277·13-s + 1/4·16-s − 0.223·20-s + 1.25·23-s + 1/5·25-s − 0.196·26-s + 1.11·29-s + 0.359·31-s + 0.176·32-s − 0.328·37-s − 0.158·40-s + 0.312·41-s + 1.21·43-s + 0.884·46-s + 1.75·47-s − 49-s + 0.141·50-s − 0.138·52-s − 1.09·53-s + 0.787·58-s − 1.04·59-s − 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.273958614\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.273958614\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−12.46075114021468, −12.32178776512737, −11.72817040904271, −11.29431310968530, −10.77230494347368, −10.53945693812269, −9.946300877435492, −9.308777239641373, −8.930081164699661, −8.480322095417704, −7.694038621710562, −7.574916481036494, −7.008023433076337, −6.456781704765749, −6.032178506775334, −5.544294414605274, −4.788581309140293, −4.615234372815348, −4.158401751454573, −3.279241844798473, −3.104604592143409, −2.514281159079361, −1.814185124880267, −1.109704204963640, −0.5124771120441508,
0.5124771120441508, 1.109704204963640, 1.814185124880267, 2.514281159079361, 3.104604592143409, 3.279241844798473, 4.158401751454573, 4.615234372815348, 4.788581309140293, 5.544294414605274, 6.032178506775334, 6.456781704765749, 7.008023433076337, 7.574916481036494, 7.694038621710562, 8.480322095417704, 8.930081164699661, 9.308777239641373, 9.946300877435492, 10.53945693812269, 10.77230494347368, 11.29431310968530, 11.72817040904271, 12.32178776512737, 12.46075114021468
