| L(s) = 1 | − 2-s + 4-s + 1.41·5-s + 1.73·7-s − 8-s + 9-s − 1.41·10-s − 1.93·11-s − 0.517·13-s − 1.73·14-s + 16-s − 17-s − 18-s + 1.41·20-s + 1.93·22-s + 23-s + 1.00·25-s + 0.517·26-s + 1.73·28-s − 32-s + 34-s + 2.44·35-s + 36-s + 0.517·37-s − 1.41·40-s − 1.93·44-s + 1.41·45-s + ⋯ |
| L(s) = 1 | − 2-s + 4-s + 1.41·5-s + 1.73·7-s − 8-s + 9-s − 1.41·10-s − 1.93·11-s − 0.517·13-s − 1.73·14-s + 16-s − 17-s − 18-s + 1.41·20-s + 1.93·22-s + 23-s + 1.00·25-s + 0.517·26-s + 1.73·28-s − 32-s + 34-s + 2.44·35-s + 36-s + 0.517·37-s − 1.41·40-s − 1.93·44-s + 1.41·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3064 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3064 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.222682764\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.222682764\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 383 | \( 1 + T \) |
| good | 3 | \( 1 - T^{2} \) |
| 5 | \( 1 - 1.41T + T^{2} \) |
| 7 | \( 1 - 1.73T + T^{2} \) |
| 11 | \( 1 + 1.93T + T^{2} \) |
| 13 | \( 1 + 0.517T + T^{2} \) |
| 17 | \( 1 + T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 - 0.517T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - 1.93T + T^{2} \) |
| 59 | \( 1 + 0.517T + T^{2} \) |
| 61 | \( 1 - 1.93T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( 1 + 1.73T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + 1.41T + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−8.807556435025806175096703479913, −8.263878467631554439860455611020, −7.40054420188427353979862920794, −7.00802202080679209499220876378, −5.76686884674470654024491473238, −5.22299554017800688738690612164, −4.48509537303349099438372574139, −2.60570414709831191378844459103, −2.17335525453623987195527974112, −1.28315310421086476249424291202,
1.28315310421086476249424291202, 2.17335525453623987195527974112, 2.60570414709831191378844459103, 4.48509537303349099438372574139, 5.22299554017800688738690612164, 5.76686884674470654024491473238, 7.00802202080679209499220876378, 7.40054420188427353979862920794, 8.263878467631554439860455611020, 8.807556435025806175096703479913
