| L(s) = 1 | + 1.30·2-s − 0.302·4-s + 0.697·7-s − 3·8-s − 1.69·11-s + 3.30·13-s + 0.908·14-s − 3.30·16-s + 1.30·17-s + 7.21·19-s − 2.21·22-s − 3.90·23-s + 4.30·26-s − 0.211·28-s + 8.60·29-s − 6.21·31-s + 1.69·32-s + 1.69·34-s + 8.90·37-s + 9.39·38-s + 1.69·41-s + 9.30·43-s + 0.513·44-s − 5.09·46-s + 8.21·47-s − 6.51·49-s − 1.00·52-s + ⋯ |
| L(s) = 1 | + 0.921·2-s − 0.151·4-s + 0.263·7-s − 1.06·8-s − 0.511·11-s + 0.916·13-s + 0.242·14-s − 0.825·16-s + 0.315·17-s + 1.65·19-s − 0.471·22-s − 0.814·23-s + 0.843·26-s − 0.0398·28-s + 1.59·29-s − 1.11·31-s + 0.300·32-s + 0.291·34-s + 1.46·37-s + 1.52·38-s + 0.265·41-s + 1.41·43-s + 0.0774·44-s − 0.750·46-s + 1.19·47-s − 0.930·49-s − 0.138·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.524776829\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.524776829\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 1.30T + 2T^{2} \) |
| 7 | \( 1 - 0.697T + 7T^{2} \) |
| 11 | \( 1 + 1.69T + 11T^{2} \) |
| 13 | \( 1 - 3.30T + 13T^{2} \) |
| 17 | \( 1 - 1.30T + 17T^{2} \) |
| 19 | \( 1 - 7.21T + 19T^{2} \) |
| 23 | \( 1 + 3.90T + 23T^{2} \) |
| 29 | \( 1 - 8.60T + 29T^{2} \) |
| 31 | \( 1 + 6.21T + 31T^{2} \) |
| 37 | \( 1 - 8.90T + 37T^{2} \) |
| 41 | \( 1 - 1.69T + 41T^{2} \) |
| 43 | \( 1 - 9.30T + 43T^{2} \) |
| 47 | \( 1 - 8.21T + 47T^{2} \) |
| 53 | \( 1 + 12.5T + 53T^{2} \) |
| 59 | \( 1 - 7.30T + 59T^{2} \) |
| 61 | \( 1 - 3.30T + 61T^{2} \) |
| 67 | \( 1 - 1.60T + 67T^{2} \) |
| 71 | \( 1 - 11.6T + 71T^{2} \) |
| 73 | \( 1 - 2.39T + 73T^{2} \) |
| 79 | \( 1 - 4.60T + 79T^{2} \) |
| 83 | \( 1 - 9T + 83T^{2} \) |
| 89 | \( 1 + 12T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−9.239237519441293871521255342653, −8.238573134005323466166479541866, −7.68366198725363219330179282207, −6.51159430598931659555008784983, −5.74617363931539656087311281631, −5.15194569520131981927564984877, −4.24039295146867036339107385486, −3.45763615800027137396861171501, −2.56327831467607934786761117001, −0.958443470869938674406594528062,
0.958443470869938674406594528062, 2.56327831467607934786761117001, 3.45763615800027137396861171501, 4.24039295146867036339107385486, 5.15194569520131981927564984877, 5.74617363931539656087311281631, 6.51159430598931659555008784983, 7.68366198725363219330179282207, 8.238573134005323466166479541866, 9.239237519441293871521255342653
