| 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
| \(L(1)\) |
\(\approx\) |
\(3.889621570\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.889621570\) |
| \(L(\frac{3}{2})\) |
|
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
−8.247484428984212621825204547599, −6.89065912953927127660851904253, −6.47169280768244603410173182146, −5.72370583004904053787001885980, −5.09293352910085181657877798931, −4.16786740622594966390002182140, −3.87482163340281587110341926541, −3.05573618344483738480354379221, −1.77778342271174657355873036800, −0.949820026495300620922184595850,
0.949820026495300620922184595850, 1.77778342271174657355873036800, 3.05573618344483738480354379221, 3.87482163340281587110341926541, 4.16786740622594966390002182140, 5.09293352910085181657877798931, 5.72370583004904053787001885980, 6.47169280768244603410173182146, 6.89065912953927127660851904253, 8.247484428984212621825204547599
