| L(s) = 1 | − 3-s − 2·5-s + 7-s + 9-s + 6·13-s + 2·15-s + 17-s − 6·19-s − 21-s + 6·23-s − 25-s − 27-s − 29-s + 8·31-s − 2·35-s − 4·37-s − 6·39-s + 3·41-s + 8·43-s − 2·45-s − 9·47-s − 6·49-s − 51-s − 53-s + 6·57-s − 11·59-s + 2·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 0.377·7-s + 1/3·9-s + 1.66·13-s + 0.516·15-s + 0.242·17-s − 1.37·19-s − 0.218·21-s + 1.25·23-s − 1/5·25-s − 0.192·27-s − 0.185·29-s + 1.43·31-s − 0.338·35-s − 0.657·37-s − 0.960·39-s + 0.468·41-s + 1.21·43-s − 0.298·45-s − 1.31·47-s − 6/7·49-s − 0.140·51-s − 0.137·53-s + 0.794·57-s − 1.43·59-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.287383792\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.287383792\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + 11 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 5 T + p T^{2} \) |
| 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
−13.77837603843493, −13.10960169262569, −12.79945136379667, −12.31800288278978, −11.55230925672147, −11.40748384439306, −10.95431208177309, −10.47968263837746, −9.993374688064373, −9.181439393329327, −8.585679723946880, −8.432345556026987, −7.712457385456765, −7.274382851173365, −6.618378632643195, −6.032695034969727, −5.838696275184692, −4.780187569599537, −4.538296972063142, −3.981958623087529, −3.319806707743438, −2.795637809669786, −1.693886981756025, −1.259314627743803, −0.3998180213840550,
0.3998180213840550, 1.259314627743803, 1.693886981756025, 2.795637809669786, 3.319806707743438, 3.981958623087529, 4.538296972063142, 4.780187569599537, 5.838696275184692, 6.032695034969727, 6.618378632643195, 7.274382851173365, 7.712457385456765, 8.432345556026987, 8.585679723946880, 9.181439393329327, 9.993374688064373, 10.47968263837746, 10.95431208177309, 11.40748384439306, 11.55230925672147, 12.31800288278978, 12.79945136379667, 13.10960169262569, 13.77837603843493
