| L(s) = 1 | − 5-s − 4·7-s + 2·13-s + 2·17-s + 4·19-s + 4·23-s + 25-s − 2·29-s + 8·31-s + 4·35-s + 6·37-s − 6·41-s − 8·43-s + 4·47-s + 9·49-s − 6·53-s − 4·59-s + 2·61-s − 2·65-s − 8·67-s + 6·73-s + 16·83-s − 2·85-s + 6·89-s − 8·91-s − 4·95-s − 14·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.51·7-s + 0.554·13-s + 0.485·17-s + 0.917·19-s + 0.834·23-s + 1/5·25-s − 0.371·29-s + 1.43·31-s + 0.676·35-s + 0.986·37-s − 0.937·41-s − 1.21·43-s + 0.583·47-s + 9/7·49-s − 0.824·53-s − 0.520·59-s + 0.256·61-s − 0.248·65-s − 0.977·67-s + 0.702·73-s + 1.75·83-s − 0.216·85-s + 0.635·89-s − 0.838·91-s − 0.410·95-s − 1.42·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−13.94306879586853, −13.56163920042792, −13.25248478922044, −12.66142932875444, −12.18169884617069, −11.75587379515290, −11.25620010428714, −10.58224501342843, −10.17198285817815, −9.587976212676915, −9.277391949715715, −8.686120556285235, −8.002159168938564, −7.636673624937010, −6.920632102010427, −6.456496529355218, −6.141884399140902, −5.302304492626751, −4.903275974593733, −4.011948906461707, −3.573513389282993, −3.005677247663989, −2.651121524808239, −1.462746888519703, −0.8402885251702993, 0,
0.8402885251702993, 1.462746888519703, 2.651121524808239, 3.005677247663989, 3.573513389282993, 4.011948906461707, 4.903275974593733, 5.302304492626751, 6.141884399140902, 6.456496529355218, 6.920632102010427, 7.636673624937010, 8.002159168938564, 8.686120556285235, 9.277391949715715, 9.587976212676915, 10.17198285817815, 10.58224501342843, 11.25620010428714, 11.75587379515290, 12.18169884617069, 12.66142932875444, 13.25248478922044, 13.56163920042792, 13.94306879586853
