L(s) = 1 | + 3-s − 3.46·5-s − 7-s + 9-s − 2·13-s − 3.46·15-s + 3.46·17-s − 19-s − 21-s + 4·23-s + 6.99·25-s + 27-s − 7.46·29-s − 2.92·31-s + 3.46·35-s − 2·37-s − 2·39-s + 6·41-s − 6.92·43-s − 3.46·45-s + 2.53·47-s + 49-s + 3.46·51-s + 3.46·53-s − 57-s + 4·59-s + 6·61-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 1.54·5-s − 0.377·7-s + 0.333·9-s − 0.554·13-s − 0.894·15-s + 0.840·17-s − 0.229·19-s − 0.218·21-s + 0.834·23-s + 1.39·25-s + 0.192·27-s − 1.38·29-s − 0.525·31-s + 0.585·35-s − 0.328·37-s − 0.320·39-s + 0.937·41-s − 1.05·43-s − 0.516·45-s + 0.369·47-s + 0.142·49-s + 0.485·51-s + 0.475·53-s − 0.132·57-s + 0.520·59-s + 0.768·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.344425676\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.344425676\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 19 | \( 1 + T \) |
good | 5 | \( 1 + 3.46T + 5T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 7.46T + 29T^{2} \) |
| 31 | \( 1 + 2.92T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 6.92T + 43T^{2} \) |
| 47 | \( 1 - 2.53T + 47T^{2} \) |
| 53 | \( 1 - 3.46T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 - 1.07T + 67T^{2} \) |
| 71 | \( 1 - 6.53T + 71T^{2} \) |
| 73 | \( 1 - 12.9T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 1.46T + 83T^{2} \) |
| 89 | \( 1 - 11.8T + 89T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
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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
−8.606470007460948377155255754626, −7.80061737834651605205031952013, −7.40308930132317026689638289573, −6.71105320129048704224351528357, −5.50527706997139096017408984842, −4.66686104195702559700868584536, −3.69838555790991062202755578297, −3.36329999769675379576435633807, −2.19726903968925207686209662830, −0.66285686493655192158641314752,
0.66285686493655192158641314752, 2.19726903968925207686209662830, 3.36329999769675379576435633807, 3.69838555790991062202755578297, 4.66686104195702559700868584536, 5.50527706997139096017408984842, 6.71105320129048704224351528357, 7.40308930132317026689638289573, 7.80061737834651605205031952013, 8.606470007460948377155255754626