| L(s) = 1 | − 1.33·2-s − 0.223·4-s + 2.26·7-s + 2.96·8-s − 4.88·11-s + 6.07·13-s − 3.01·14-s − 3.50·16-s − 4.32·17-s − 7.22·19-s + 6.51·22-s − 4.12·23-s − 8.10·26-s − 0.504·28-s − 4.04·29-s − 31-s − 1.25·32-s + 5.76·34-s − 6.36·37-s + 9.63·38-s + 8.39·41-s + 5.73·43-s + 1.09·44-s + 5.50·46-s − 10.7·47-s − 1.88·49-s − 1.35·52-s + ⋯ |
| L(s) = 1 | − 0.942·2-s − 0.111·4-s + 0.855·7-s + 1.04·8-s − 1.47·11-s + 1.68·13-s − 0.805·14-s − 0.876·16-s − 1.04·17-s − 1.65·19-s + 1.38·22-s − 0.860·23-s − 1.58·26-s − 0.0953·28-s − 0.750·29-s − 0.179·31-s − 0.221·32-s + 0.989·34-s − 1.04·37-s + 1.56·38-s + 1.31·41-s + 0.874·43-s + 0.164·44-s + 0.811·46-s − 1.56·47-s − 0.268·49-s − 0.187·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7593888964\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7593888964\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 + 1.33T + 2T^{2} \) |
| 7 | \( 1 - 2.26T + 7T^{2} \) |
| 11 | \( 1 + 4.88T + 11T^{2} \) |
| 13 | \( 1 - 6.07T + 13T^{2} \) |
| 17 | \( 1 + 4.32T + 17T^{2} \) |
| 19 | \( 1 + 7.22T + 19T^{2} \) |
| 23 | \( 1 + 4.12T + 23T^{2} \) |
| 29 | \( 1 + 4.04T + 29T^{2} \) |
| 37 | \( 1 + 6.36T + 37T^{2} \) |
| 41 | \( 1 - 8.39T + 41T^{2} \) |
| 43 | \( 1 - 5.73T + 43T^{2} \) |
| 47 | \( 1 + 10.7T + 47T^{2} \) |
| 53 | \( 1 - 9.22T + 53T^{2} \) |
| 59 | \( 1 - 0.990T + 59T^{2} \) |
| 61 | \( 1 - 10.2T + 61T^{2} \) |
| 67 | \( 1 - 11.6T + 67T^{2} \) |
| 71 | \( 1 + 9.62T + 71T^{2} \) |
| 73 | \( 1 + 1.35T + 73T^{2} \) |
| 79 | \( 1 - 0.231T + 79T^{2} \) |
| 83 | \( 1 - 4.96T + 83T^{2} \) |
| 89 | \( 1 - 17.8T + 89T^{2} \) |
| 97 | \( 1 + 8.43T + 97T^{2} \) |
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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
−8.144836904877996266975082230398, −7.59357206172534682704529406401, −6.67647773975755664071428864855, −5.88398907283139353629970743218, −5.09660780501182681368923046276, −4.33918513014302817336931415112, −3.73836053068419452174246004752, −2.28284461359980636706485100212, −1.77070856321670393313696205609, −0.51113320151512273896084321759,
0.51113320151512273896084321759, 1.77070856321670393313696205609, 2.28284461359980636706485100212, 3.73836053068419452174246004752, 4.33918513014302817336931415112, 5.09660780501182681368923046276, 5.88398907283139353629970743218, 6.67647773975755664071428864855, 7.59357206172534682704529406401, 8.144836904877996266975082230398
