| L(s) = 1 | + 2-s + 4-s + 4.41·7-s + 8-s + 1.41·11-s + 5.82·13-s + 4.41·14-s + 16-s − 17-s − 19-s + 1.41·22-s − 0.757·23-s + 5.82·26-s + 4.41·28-s + 0.171·29-s + 6.24·31-s + 32-s − 34-s − 8.48·37-s − 38-s + 4.24·41-s + 1.75·43-s + 1.41·44-s − 0.757·46-s + 12.4·49-s + 5.82·52-s − 5.48·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.66·7-s + 0.353·8-s + 0.426·11-s + 1.61·13-s + 1.17·14-s + 0.250·16-s − 0.242·17-s − 0.229·19-s + 0.301·22-s − 0.157·23-s + 1.14·26-s + 0.834·28-s + 0.0318·29-s + 1.12·31-s + 0.176·32-s − 0.171·34-s − 1.39·37-s − 0.162·38-s + 0.662·41-s + 0.267·43-s + 0.213·44-s − 0.111·46-s + 1.78·49-s + 0.808·52-s − 0.753·53-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
| \(L(1)\) |
\(\approx\) |
\(4.966225998\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.966225998\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 4.41T + 7T^{2} \) |
| 11 | \( 1 - 1.41T + 11T^{2} \) |
| 13 | \( 1 - 5.82T + 13T^{2} \) |
| 17 | \( 1 + T + 17T^{2} \) |
| 23 | \( 1 + 0.757T + 23T^{2} \) |
| 29 | \( 1 - 0.171T + 29T^{2} \) |
| 31 | \( 1 - 6.24T + 31T^{2} \) |
| 37 | \( 1 + 8.48T + 37T^{2} \) |
| 41 | \( 1 - 4.24T + 41T^{2} \) |
| 43 | \( 1 - 1.75T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 5.48T + 53T^{2} \) |
| 59 | \( 1 + 6.89T + 59T^{2} \) |
| 61 | \( 1 - 14.2T + 61T^{2} \) |
| 67 | \( 1 - 4.75T + 67T^{2} \) |
| 71 | \( 1 - 13.4T + 71T^{2} \) |
| 73 | \( 1 - 11.4T + 73T^{2} \) |
| 79 | \( 1 + 6.48T + 79T^{2} \) |
| 83 | \( 1 + 14.4T + 83T^{2} \) |
| 89 | \( 1 + 7.07T + 89T^{2} \) |
| 97 | \( 1 + 0.343T + 97T^{2} \) |
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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
−7.916198572879353642813300380482, −6.90887708408965047979866863337, −6.38626063195253612822281471128, −5.55814047309776613225682615586, −5.01576708376672632726759356209, −4.17580433616505669081611166662, −3.78585933314799342505132112432, −2.65624352072039710110842186887, −1.72455973947737535497831794449, −1.10142381880238312606238711056,
1.10142381880238312606238711056, 1.72455973947737535497831794449, 2.65624352072039710110842186887, 3.78585933314799342505132112432, 4.17580433616505669081611166662, 5.01576708376672632726759356209, 5.55814047309776613225682615586, 6.38626063195253612822281471128, 6.90887708408965047979866863337, 7.916198572879353642813300380482
