| L(s) = 1 | + 0.728·2-s − 1.46·4-s − 2.55·7-s − 2.52·8-s − 2.46·11-s − 1.55·13-s − 1.86·14-s + 1.09·16-s − 6.83·17-s − 7.66·19-s − 1.79·22-s − 7.50·23-s − 1.13·26-s + 3.74·28-s − 3.25·29-s + 0.658·31-s + 5.85·32-s − 4.98·34-s − 37-s − 5.58·38-s − 2.46·41-s − 10.9·43-s + 3.62·44-s − 5.46·46-s + 3.11·47-s − 0.485·49-s + 2.28·52-s + ⋯ |
| L(s) = 1 | + 0.515·2-s − 0.734·4-s − 0.964·7-s − 0.893·8-s − 0.744·11-s − 0.432·13-s − 0.497·14-s + 0.273·16-s − 1.65·17-s − 1.75·19-s − 0.383·22-s − 1.56·23-s − 0.222·26-s + 0.708·28-s − 0.604·29-s + 0.118·31-s + 1.03·32-s − 0.854·34-s − 0.164·37-s − 0.905·38-s − 0.385·41-s − 1.67·43-s + 0.546·44-s − 0.806·46-s + 0.454·47-s − 0.0692·49-s + 0.317·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1112761496\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1112761496\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 2 | \( 1 - 0.728T + 2T^{2} \) |
| 7 | \( 1 + 2.55T + 7T^{2} \) |
| 11 | \( 1 + 2.46T + 11T^{2} \) |
| 13 | \( 1 + 1.55T + 13T^{2} \) |
| 17 | \( 1 + 6.83T + 17T^{2} \) |
| 19 | \( 1 + 7.66T + 19T^{2} \) |
| 23 | \( 1 + 7.50T + 23T^{2} \) |
| 29 | \( 1 + 3.25T + 29T^{2} \) |
| 31 | \( 1 - 0.658T + 31T^{2} \) |
| 41 | \( 1 + 2.46T + 41T^{2} \) |
| 43 | \( 1 + 10.9T + 43T^{2} \) |
| 47 | \( 1 - 3.11T + 47T^{2} \) |
| 53 | \( 1 - 8.64T + 53T^{2} \) |
| 59 | \( 1 - 6.23T + 59T^{2} \) |
| 61 | \( 1 - 3.27T + 61T^{2} \) |
| 67 | \( 1 + 1.47T + 67T^{2} \) |
| 71 | \( 1 - 8.06T + 71T^{2} \) |
| 73 | \( 1 - 4.96T + 73T^{2} \) |
| 79 | \( 1 - 12.8T + 79T^{2} \) |
| 83 | \( 1 - 1.14T + 83T^{2} \) |
| 89 | \( 1 + 11.5T + 89T^{2} \) |
| 97 | \( 1 + 17.2T + 97T^{2} \) |
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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
−7.994467622047032627789255804243, −6.72245937035050925923374523093, −6.53438612506067369606893250147, −5.62909283408869987785370640412, −4.99113420850431625300713811761, −4.12694876365219252011961791404, −3.79719806108412711039889575350, −2.66876503555047697027943861224, −2.07495220602407452694949807295, −0.13942865302441229394573967887,
0.13942865302441229394573967887, 2.07495220602407452694949807295, 2.66876503555047697027943861224, 3.79719806108412711039889575350, 4.12694876365219252011961791404, 4.99113420850431625300713811761, 5.62909283408869987785370640412, 6.53438612506067369606893250147, 6.72245937035050925923374523093, 7.994467622047032627789255804243
