L(s) = 1 | − 2-s + 4-s − 4.42·7-s − 8-s − 2.62·11-s + 5.80·13-s + 4.42·14-s + 16-s + 3.80·17-s + 19-s + 2.62·22-s + 2.62·23-s − 5.80·26-s − 4.42·28-s − 3.37·29-s − 4.42·31-s − 32-s − 3.80·34-s + 5.80·37-s − 38-s − 5.67·41-s − 10.9·43-s − 2.62·44-s − 2.62·46-s − 2.62·47-s + 12.6·49-s + 5.80·52-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.67·7-s − 0.353·8-s − 0.790·11-s + 1.61·13-s + 1.18·14-s + 0.250·16-s + 0.923·17-s + 0.229·19-s + 0.559·22-s + 0.546·23-s − 1.13·26-s − 0.836·28-s − 0.627·29-s − 0.795·31-s − 0.176·32-s − 0.652·34-s + 0.954·37-s − 0.162·38-s − 0.885·41-s − 1.67·43-s − 0.395·44-s − 0.386·46-s − 0.382·47-s + 1.80·49-s + 0.805·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9337983801\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9337983801\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 7 | \( 1 + 4.42T + 7T^{2} \) |
| 11 | \( 1 + 2.62T + 11T^{2} \) |
| 13 | \( 1 - 5.80T + 13T^{2} \) |
| 17 | \( 1 - 3.80T + 17T^{2} \) |
| 23 | \( 1 - 2.62T + 23T^{2} \) |
| 29 | \( 1 + 3.37T + 29T^{2} \) |
| 31 | \( 1 + 4.42T + 31T^{2} \) |
| 37 | \( 1 - 5.80T + 37T^{2} \) |
| 41 | \( 1 + 5.67T + 41T^{2} \) |
| 43 | \( 1 + 10.9T + 43T^{2} \) |
| 47 | \( 1 + 2.62T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 1.05T + 59T^{2} \) |
| 61 | \( 1 - 4.75T + 61T^{2} \) |
| 67 | \( 1 - 15.6T + 67T^{2} \) |
| 71 | \( 1 + 15.6T + 71T^{2} \) |
| 73 | \( 1 - 11.6T + 73T^{2} \) |
| 79 | \( 1 - 4.42T + 79T^{2} \) |
| 83 | \( 1 + 11.9T + 83T^{2} \) |
| 89 | \( 1 + 12.4T + 89T^{2} \) |
| 97 | \( 1 - 7.37T + 97T^{2} \) |
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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
−7.912643469923216129853748288153, −6.96868488315335213408946055544, −6.62487777698329249771059391991, −5.75643926846827802732727431159, −5.36046734424467050629469557726, −3.90853117734720178101251036931, −3.35943424569274292192689227386, −2.76576253754938955573947207686, −1.56373350943480798128875077943, −0.53372213282083449528426349315,
0.53372213282083449528426349315, 1.56373350943480798128875077943, 2.76576253754938955573947207686, 3.35943424569274292192689227386, 3.90853117734720178101251036931, 5.36046734424467050629469557726, 5.75643926846827802732727431159, 6.62487777698329249771059391991, 6.96868488315335213408946055544, 7.912643469923216129853748288153