L(s) = 1 | − 2·49-s + 4·73-s − 81-s + 4·97-s − 4·103-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯ |
L(s) = 1 | − 2·49-s + 4·73-s − 81-s + 4·97-s − 4·103-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.367900537\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.367900537\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.37422552127631811990744904537, −6.07003763132055325354077506711, −6.04811878382530264620116410581, −5.64060467304776256336655797492, −5.33667137630832365099942128043, −5.27409178840062880015855129755, −5.19414552198861979278733770581, −4.99406100000956186268089200398, −4.54888044872619501393177881343, −4.49038104802090756636012496738, −4.36764642988661584798371395177, −4.00508583512675145322602768476, −3.68317865678735530973936744716, −3.68271564241593316547300509276, −3.46433824453624529971639297433, −3.04580474335413517861486799073, −2.86572896875207406676855758742, −2.84176195310126380926635618530, −2.30594605071210871582426499340, −2.16157231065177360550707366089, −1.87219009593183543242796165403, −1.69352324724406189620353224899, −1.24973401361455021484305002856, −0.973027253471665838444414465238, −0.48976174427291262581174966561,
0.48976174427291262581174966561, 0.973027253471665838444414465238, 1.24973401361455021484305002856, 1.69352324724406189620353224899, 1.87219009593183543242796165403, 2.16157231065177360550707366089, 2.30594605071210871582426499340, 2.84176195310126380926635618530, 2.86572896875207406676855758742, 3.04580474335413517861486799073, 3.46433824453624529971639297433, 3.68271564241593316547300509276, 3.68317865678735530973936744716, 4.00508583512675145322602768476, 4.36764642988661584798371395177, 4.49038104802090756636012496738, 4.54888044872619501393177881343, 4.99406100000956186268089200398, 5.19414552198861979278733770581, 5.27409178840062880015855129755, 5.33667137630832365099942128043, 5.64060467304776256336655797492, 6.04811878382530264620116410581, 6.07003763132055325354077506711, 6.37422552127631811990744904537