| L(s) = 1 | + 3-s − 1.15·5-s − 7-s + 9-s − 3.66·11-s + 2.34·13-s − 1.15·15-s + 4.80·17-s − 7.06·19-s − 21-s + 23-s − 3.66·25-s + 27-s − 6.52·29-s − 2.80·31-s − 3.66·33-s + 1.15·35-s + 5.40·37-s + 2.34·39-s + 0.647·41-s + 4.76·43-s − 1.15·45-s − 4.85·47-s + 49-s + 4.80·51-s + 7.63·53-s + 4.24·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.517·5-s − 0.377·7-s + 0.333·9-s − 1.10·11-s + 0.650·13-s − 0.298·15-s + 1.16·17-s − 1.62·19-s − 0.218·21-s + 0.208·23-s − 0.732·25-s + 0.192·27-s − 1.21·29-s − 0.504·31-s − 0.638·33-s + 0.195·35-s + 0.889·37-s + 0.375·39-s + 0.101·41-s + 0.726·43-s − 0.172·45-s − 0.708·47-s + 0.142·49-s + 0.673·51-s + 1.04·53-s + 0.572·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.700489346\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.700489346\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + 1.15T + 5T^{2} \) |
| 11 | \( 1 + 3.66T + 11T^{2} \) |
| 13 | \( 1 - 2.34T + 13T^{2} \) |
| 17 | \( 1 - 4.80T + 17T^{2} \) |
| 19 | \( 1 + 7.06T + 19T^{2} \) |
| 29 | \( 1 + 6.52T + 29T^{2} \) |
| 31 | \( 1 + 2.80T + 31T^{2} \) |
| 37 | \( 1 - 5.40T + 37T^{2} \) |
| 41 | \( 1 - 0.647T + 41T^{2} \) |
| 43 | \( 1 - 4.76T + 43T^{2} \) |
| 47 | \( 1 + 4.85T + 47T^{2} \) |
| 53 | \( 1 - 7.63T + 53T^{2} \) |
| 59 | \( 1 - 10.2T + 59T^{2} \) |
| 61 | \( 1 - 9.91T + 61T^{2} \) |
| 67 | \( 1 + 6.07T + 67T^{2} \) |
| 71 | \( 1 - 3.65T + 71T^{2} \) |
| 73 | \( 1 - 11.8T + 73T^{2} \) |
| 79 | \( 1 - 9.12T + 79T^{2} \) |
| 83 | \( 1 - 15.8T + 83T^{2} \) |
| 89 | \( 1 - 4.66T + 89T^{2} \) |
| 97 | \( 1 + 4.90T + 97T^{2} \) |
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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
−7.966872152009392232675484051988, −7.36300004482275365262004312502, −6.52935943276771885537623216891, −5.75541879744627958753798523512, −5.10244211650645082632408141066, −3.94081253444707000488836023952, −3.71933069123949817703604367639, −2.67615953343853287785242526823, −1.97009030097406763229838553370, −0.61090269437896549514952388492,
0.61090269437896549514952388492, 1.97009030097406763229838553370, 2.67615953343853287785242526823, 3.71933069123949817703604367639, 3.94081253444707000488836023952, 5.10244211650645082632408141066, 5.75541879744627958753798523512, 6.52935943276771885537623216891, 7.36300004482275365262004312502, 7.966872152009392232675484051988
