| L(s) = 1 | − 2-s + 4-s + 4.44·7-s − 8-s + 1.44·11-s − 2.44·13-s − 4.44·14-s + 16-s + 3.89·17-s − 0.550·19-s − 1.44·22-s − 2.89·23-s + 2.44·26-s + 4.44·28-s − 6·29-s + 6.44·31-s − 32-s − 3.89·34-s + 8·37-s + 0.550·38-s − 41-s + 7.44·43-s + 1.44·44-s + 2.89·46-s + 0.449·47-s + 12.7·49-s − 2.44·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.68·7-s − 0.353·8-s + 0.437·11-s − 0.679·13-s − 1.18·14-s + 0.250·16-s + 0.945·17-s − 0.126·19-s − 0.309·22-s − 0.604·23-s + 0.480·26-s + 0.840·28-s − 1.11·29-s + 1.15·31-s − 0.176·32-s − 0.668·34-s + 1.31·37-s + 0.0893·38-s − 0.156·41-s + 1.13·43-s + 0.218·44-s + 0.427·46-s + 0.0655·47-s + 1.82·49-s − 0.339·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.780093107\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.780093107\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 - 4.44T + 7T^{2} \) |
| 11 | \( 1 - 1.44T + 11T^{2} \) |
| 13 | \( 1 + 2.44T + 13T^{2} \) |
| 17 | \( 1 - 3.89T + 17T^{2} \) |
| 19 | \( 1 + 0.550T + 19T^{2} \) |
| 23 | \( 1 + 2.89T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 6.44T + 31T^{2} \) |
| 37 | \( 1 - 8T + 37T^{2} \) |
| 41 | \( 1 + T + 41T^{2} \) |
| 43 | \( 1 - 7.44T + 43T^{2} \) |
| 47 | \( 1 - 0.449T + 47T^{2} \) |
| 53 | \( 1 - 8.44T + 53T^{2} \) |
| 59 | \( 1 + 11.2T + 59T^{2} \) |
| 61 | \( 1 + 0.449T + 61T^{2} \) |
| 67 | \( 1 - 9.44T + 67T^{2} \) |
| 71 | \( 1 - 2.44T + 71T^{2} \) |
| 73 | \( 1 + 4.79T + 73T^{2} \) |
| 79 | \( 1 + 7.34T + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 - 12.8T + 89T^{2} \) |
| 97 | \( 1 - 13T + 97T^{2} \) |
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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
−8.339563986932870424202180468562, −7.72382008668861307350916612986, −7.38215223168032834681833095580, −6.24779322808817161528656473236, −5.51590793892054510298416854158, −4.69231405209101975306355702254, −3.92371181927958158556053464890, −2.64139178383898371260436569300, −1.80484528932071689015137879459, −0.896780737417836870195025678475,
0.896780737417836870195025678475, 1.80484528932071689015137879459, 2.64139178383898371260436569300, 3.92371181927958158556053464890, 4.69231405209101975306355702254, 5.51590793892054510298416854158, 6.24779322808817161528656473236, 7.38215223168032834681833095580, 7.72382008668861307350916612986, 8.339563986932870424202180468562
