| L(s) = 1 | − 0.445·3-s + 4-s − 1.80·5-s − 0.801·9-s + 11-s − 0.445·12-s + 0.801·15-s + 16-s − 1.80·20-s + 1.24·23-s + 2.24·25-s + 0.801·27-s + 1.24·31-s − 0.445·33-s − 0.801·36-s − 0.445·37-s + 44-s + 1.44·45-s + 1.24·47-s − 0.445·48-s + 49-s − 1.80·53-s − 1.80·55-s − 0.445·59-s + 0.801·60-s + 64-s + 2·67-s + ⋯ |
| L(s) = 1 | − 0.445·3-s + 4-s − 1.80·5-s − 0.801·9-s + 11-s − 0.445·12-s + 0.801·15-s + 16-s − 1.80·20-s + 1.24·23-s + 2.24·25-s + 0.801·27-s + 1.24·31-s − 0.445·33-s − 0.801·36-s − 0.445·37-s + 44-s + 1.44·45-s + 1.24·47-s − 0.445·48-s + 49-s − 1.80·53-s − 1.80·55-s − 0.445·59-s + 0.801·60-s + 64-s + 2·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9161278518\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9161278518\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - T^{2} \) |
| 3 | \( 1 + 0.445T + T^{2} \) |
| 5 | \( 1 + 1.80T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - 1.24T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - 1.24T + T^{2} \) |
| 37 | \( 1 + 0.445T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - 1.24T + T^{2} \) |
| 53 | \( 1 + 1.80T + T^{2} \) |
| 59 | \( 1 + 0.445T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - 2T + T^{2} \) |
| 71 | \( 1 + 0.445T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - 1.24T + T^{2} \) |
| 97 | \( 1 + 1.80T + T^{2} \) |
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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
−9.292056194875877326632740267655, −8.441850146196886727752900434290, −7.84251593840551186637965948423, −6.94919723085321288147877295584, −6.52550399761612344959507505872, −5.43136315875746284702129437035, −4.41904343795872049697922612782, −3.50146483921704539811110885925, −2.75153906743668587167462864341, −1.00581983433485497460450871395,
1.00581983433485497460450871395, 2.75153906743668587167462864341, 3.50146483921704539811110885925, 4.41904343795872049697922612782, 5.43136315875746284702129437035, 6.52550399761612344959507505872, 6.94919723085321288147877295584, 7.84251593840551186637965948423, 8.441850146196886727752900434290, 9.292056194875877326632740267655
