L(s) = 1 | + 2·3-s + 3·5-s + 7-s + 9-s − 3·11-s − 4·13-s + 6·15-s − 3·17-s − 19-s + 2·21-s + 4·25-s − 4·27-s + 6·29-s + 4·31-s − 6·33-s + 3·35-s + 2·37-s − 8·39-s − 6·41-s + 43-s + 3·45-s + 3·47-s − 6·49-s − 6·51-s + 12·53-s − 9·55-s − 2·57-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 1.34·5-s + 0.377·7-s + 1/3·9-s − 0.904·11-s − 1.10·13-s + 1.54·15-s − 0.727·17-s − 0.229·19-s + 0.436·21-s + 4/5·25-s − 0.769·27-s + 1.11·29-s + 0.718·31-s − 1.04·33-s + 0.507·35-s + 0.328·37-s − 1.28·39-s − 0.937·41-s + 0.152·43-s + 0.447·45-s + 0.437·47-s − 6/7·49-s − 0.840·51-s + 1.64·53-s − 1.21·55-s − 0.264·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.063546195\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.063546195\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−11.77163153247066108695618360086, −10.40992229609183401889707132964, −9.834844837220933208096577473942, −8.867143248873807381249886076350, −8.105075898459616611364991061094, −6.97094335247891624980251070075, −5.68944505001593374643432676068, −4.62455059365688352928682233512, −2.80628438558787874241283090829, −2.10540195460897363050983106218,
2.10540195460897363050983106218, 2.80628438558787874241283090829, 4.62455059365688352928682233512, 5.68944505001593374643432676068, 6.97094335247891624980251070075, 8.105075898459616611364991061094, 8.867143248873807381249886076350, 9.834844837220933208096577473942, 10.40992229609183401889707132964, 11.77163153247066108695618360086