| L(s) = 1 | + 2-s + 2·3-s + 4-s + 5-s + 2·6-s + 1.37·7-s + 8-s + 9-s + 10-s − 3.37·11-s + 2·12-s − 4.74·13-s + 1.37·14-s + 2·15-s + 16-s − 5.37·17-s + 18-s − 2·19-s + 20-s + 2.74·21-s − 3.37·22-s + 6.74·23-s + 2·24-s + 25-s − 4.74·26-s − 4·27-s + 1.37·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s + 0.5·4-s + 0.447·5-s + 0.816·6-s + 0.518·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s − 1.01·11-s + 0.577·12-s − 1.31·13-s + 0.366·14-s + 0.516·15-s + 0.250·16-s − 1.30·17-s + 0.235·18-s − 0.458·19-s + 0.223·20-s + 0.598·21-s − 0.718·22-s + 1.40·23-s + 0.408·24-s + 0.200·25-s − 0.930·26-s − 0.769·27-s + 0.259·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.830337769\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.830337769\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 37 | \( 1 - T \) |
| good | 3 | \( 1 - 2T + 3T^{2} \) |
| 7 | \( 1 - 1.37T + 7T^{2} \) |
| 11 | \( 1 + 3.37T + 11T^{2} \) |
| 13 | \( 1 + 4.74T + 13T^{2} \) |
| 17 | \( 1 + 5.37T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 - 6.74T + 23T^{2} \) |
| 29 | \( 1 - 8.11T + 29T^{2} \) |
| 31 | \( 1 + 2.62T + 31T^{2} \) |
| 41 | \( 1 - 5.37T + 41T^{2} \) |
| 43 | \( 1 - 7.37T + 43T^{2} \) |
| 47 | \( 1 + 8.74T + 47T^{2} \) |
| 53 | \( 1 - 1.37T + 53T^{2} \) |
| 59 | \( 1 - 12.7T + 59T^{2} \) |
| 61 | \( 1 + 5.37T + 61T^{2} \) |
| 67 | \( 1 - 4.74T + 67T^{2} \) |
| 71 | \( 1 + 6.74T + 71T^{2} \) |
| 73 | \( 1 - 8.74T + 73T^{2} \) |
| 79 | \( 1 + 4.74T + 79T^{2} \) |
| 83 | \( 1 + 0.744T + 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 + 0.116T + 97T^{2} \) |
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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
−11.38722906739153977350080833803, −10.54595895376619615626499094419, −9.480271056268451581901414239945, −8.554390353285712280271927196282, −7.67557411468823720785337737584, −6.70394729365155325435215824550, −5.26930273654908337611692424533, −4.46965523456966547152859814665, −2.86056370193655894657740559744, −2.24525849682385529795497473869,
2.24525849682385529795497473869, 2.86056370193655894657740559744, 4.46965523456966547152859814665, 5.26930273654908337611692424533, 6.70394729365155325435215824550, 7.67557411468823720785337737584, 8.554390353285712280271927196282, 9.480271056268451581901414239945, 10.54595895376619615626499094419, 11.38722906739153977350080833803
