| L(s) = 1 | + 0.445·2-s − 1.80·4-s + 2.35·5-s − 3.24·7-s − 1.69·8-s + 1.04·10-s − 1.15·11-s + 2.04·13-s − 1.44·14-s + 2.85·16-s + 4.54·17-s − 2.95·19-s − 4.24·20-s − 0.515·22-s − 4.24·23-s + 0.554·25-s + 0.911·26-s + 5.85·28-s − 5.54·29-s − 10.4·31-s + 4.65·32-s + 2.02·34-s − 7.65·35-s + 0.329·37-s − 1.31·38-s − 3.98·40-s − 4.02·41-s + ⋯ |
| L(s) = 1 | + 0.314·2-s − 0.900·4-s + 1.05·5-s − 1.22·7-s − 0.598·8-s + 0.331·10-s − 0.349·11-s + 0.568·13-s − 0.386·14-s + 0.712·16-s + 1.10·17-s − 0.677·19-s − 0.949·20-s − 0.109·22-s − 0.885·23-s + 0.110·25-s + 0.178·26-s + 1.10·28-s − 1.02·29-s − 1.86·31-s + 0.822·32-s + 0.346·34-s − 1.29·35-s + 0.0542·37-s − 0.213·38-s − 0.630·40-s − 0.628·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1341 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1341 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 149 | \( 1 - T \) |
| good | 2 | \( 1 - 0.445T + 2T^{2} \) |
| 5 | \( 1 - 2.35T + 5T^{2} \) |
| 7 | \( 1 + 3.24T + 7T^{2} \) |
| 11 | \( 1 + 1.15T + 11T^{2} \) |
| 13 | \( 1 - 2.04T + 13T^{2} \) |
| 17 | \( 1 - 4.54T + 17T^{2} \) |
| 19 | \( 1 + 2.95T + 19T^{2} \) |
| 23 | \( 1 + 4.24T + 23T^{2} \) |
| 29 | \( 1 + 5.54T + 29T^{2} \) |
| 31 | \( 1 + 10.4T + 31T^{2} \) |
| 37 | \( 1 - 0.329T + 37T^{2} \) |
| 41 | \( 1 + 4.02T + 41T^{2} \) |
| 43 | \( 1 + 5.18T + 43T^{2} \) |
| 47 | \( 1 + 9.94T + 47T^{2} \) |
| 53 | \( 1 - 0.829T + 53T^{2} \) |
| 59 | \( 1 - 5.65T + 59T^{2} \) |
| 61 | \( 1 + 0.0217T + 61T^{2} \) |
| 67 | \( 1 + 9.24T + 67T^{2} \) |
| 71 | \( 1 - 12.7T + 71T^{2} \) |
| 73 | \( 1 + 14.6T + 73T^{2} \) |
| 79 | \( 1 + 6.38T + 79T^{2} \) |
| 83 | \( 1 - 0.801T + 83T^{2} \) |
| 89 | \( 1 - 8.42T + 89T^{2} \) |
| 97 | \( 1 - 18.9T + 97T^{2} \) |
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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
−9.353308522023383309135467148283, −8.616050107991238616047853852313, −7.61125962963357688510779533124, −6.40018359155286333829920804992, −5.82627930020205305386601796447, −5.18747011428428757530337858902, −3.84893798086036659130038301831, −3.23355182239828265139685651623, −1.80987494269346383518149171800, 0,
1.80987494269346383518149171800, 3.23355182239828265139685651623, 3.84893798086036659130038301831, 5.18747011428428757530337858902, 5.82627930020205305386601796447, 6.40018359155286333829920804992, 7.61125962963357688510779533124, 8.616050107991238616047853852313, 9.353308522023383309135467148283
