| L(s) = 1 | + 1.24·3-s + 2.83·5-s + 4.46·7-s − 1.45·9-s − 11-s + 2.14·13-s + 3.51·15-s + 7.81·17-s − 19-s + 5.54·21-s − 3.81·23-s + 3.02·25-s − 5.53·27-s + 1.86·29-s + 1.22·31-s − 1.24·33-s + 12.6·35-s − 8.10·37-s + 2.65·39-s + 10.9·41-s − 2.97·43-s − 4.13·45-s − 6.28·47-s + 12.9·49-s + 9.69·51-s + 6.43·53-s − 2.83·55-s + ⋯ |
| L(s) = 1 | + 0.716·3-s + 1.26·5-s + 1.68·7-s − 0.486·9-s − 0.301·11-s + 0.594·13-s + 0.907·15-s + 1.89·17-s − 0.229·19-s + 1.20·21-s − 0.794·23-s + 0.604·25-s − 1.06·27-s + 0.346·29-s + 0.220·31-s − 0.216·33-s + 2.13·35-s − 1.33·37-s + 0.425·39-s + 1.70·41-s − 0.454·43-s − 0.616·45-s − 0.916·47-s + 1.84·49-s + 1.35·51-s + 0.884·53-s − 0.381·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.815357050\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.815357050\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 1.24T + 3T^{2} \) |
| 5 | \( 1 - 2.83T + 5T^{2} \) |
| 7 | \( 1 - 4.46T + 7T^{2} \) |
| 13 | \( 1 - 2.14T + 13T^{2} \) |
| 17 | \( 1 - 7.81T + 17T^{2} \) |
| 23 | \( 1 + 3.81T + 23T^{2} \) |
| 29 | \( 1 - 1.86T + 29T^{2} \) |
| 31 | \( 1 - 1.22T + 31T^{2} \) |
| 37 | \( 1 + 8.10T + 37T^{2} \) |
| 41 | \( 1 - 10.9T + 41T^{2} \) |
| 43 | \( 1 + 2.97T + 43T^{2} \) |
| 47 | \( 1 + 6.28T + 47T^{2} \) |
| 53 | \( 1 - 6.43T + 53T^{2} \) |
| 59 | \( 1 + 7.40T + 59T^{2} \) |
| 61 | \( 1 + 9.83T + 61T^{2} \) |
| 67 | \( 1 - 13.7T + 67T^{2} \) |
| 71 | \( 1 - 2.27T + 71T^{2} \) |
| 73 | \( 1 - 2.77T + 73T^{2} \) |
| 79 | \( 1 + 15.9T + 79T^{2} \) |
| 83 | \( 1 + 6.37T + 83T^{2} \) |
| 89 | \( 1 - 1.65T + 89T^{2} \) |
| 97 | \( 1 + 5.99T + 97T^{2} \) |
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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
−8.465701520485009438991513846272, −8.077502421299344910141340915487, −7.41400287197374038318479501368, −6.11968227855861220915734710576, −5.59307781997008295682885265963, −4.98004714911563365112803505693, −3.86077193420406117378507007735, −2.85777714163330582435021888006, −1.97958108537885702832940058949, −1.29945929374827035025862930804,
1.29945929374827035025862930804, 1.97958108537885702832940058949, 2.85777714163330582435021888006, 3.86077193420406117378507007735, 4.98004714911563365112803505693, 5.59307781997008295682885265963, 6.11968227855861220915734710576, 7.41400287197374038318479501368, 8.077502421299344910141340915487, 8.465701520485009438991513846272
