L(s) = 1 | + 3-s − 5-s − 2·7-s + 9-s + 2·11-s − 15-s − 7·17-s + 6·19-s − 2·21-s + 6·23-s − 4·25-s + 27-s − 29-s − 4·31-s + 2·33-s + 2·35-s + 37-s + 9·41-s − 6·43-s − 45-s − 6·47-s − 3·49-s − 7·51-s − 9·53-s − 2·55-s + 6·57-s + 61-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s + 0.603·11-s − 0.258·15-s − 1.69·17-s + 1.37·19-s − 0.436·21-s + 1.25·23-s − 4/5·25-s + 0.192·27-s − 0.185·29-s − 0.718·31-s + 0.348·33-s + 0.338·35-s + 0.164·37-s + 1.40·41-s − 0.914·43-s − 0.149·45-s − 0.875·47-s − 3/7·49-s − 0.980·51-s − 1.23·53-s − 0.269·55-s + 0.794·57-s + 0.128·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−7.50000123702294602244531582764, −6.78678644595573045824963421641, −6.35508561457601568136320514661, −5.30380324801450203656673424389, −4.54364176101767494390618686453, −3.74311519316927651394687859176, −3.20872751880145323392672979333, −2.34679173587700529989286861745, −1.29246487757457700291564957499, 0,
1.29246487757457700291564957499, 2.34679173587700529989286861745, 3.20872751880145323392672979333, 3.74311519316927651394687859176, 4.54364176101767494390618686453, 5.30380324801450203656673424389, 6.35508561457601568136320514661, 6.78678644595573045824963421641, 7.50000123702294602244531582764