| L(s) = 1 | − 3-s − 2.64·5-s − 7-s + 9-s − 6.24·11-s + 4·13-s + 2.64·15-s − 2.64·17-s − 19-s + 21-s − 6.24·23-s + 1.96·25-s − 27-s + 3.60·29-s − 8.24·31-s + 6.24·33-s + 2.64·35-s + 2·37-s − 4·39-s − 5.21·41-s + 4.24·43-s − 2.64·45-s + 1.60·47-s + 49-s + 2.64·51-s − 12.8·53-s + 16.4·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.18·5-s − 0.377·7-s + 0.333·9-s − 1.88·11-s + 1.10·13-s + 0.681·15-s − 0.640·17-s − 0.229·19-s + 0.218·21-s − 1.30·23-s + 0.393·25-s − 0.192·27-s + 0.670·29-s − 1.48·31-s + 1.08·33-s + 0.446·35-s + 0.328·37-s − 0.640·39-s − 0.815·41-s + 0.648·43-s − 0.393·45-s + 0.234·47-s + 0.142·49-s + 0.369·51-s − 1.77·53-s + 2.22·55-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
| \(L(1)\) |
\(\approx\) |
\(0.4514471662\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4514471662\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.64T + 5T^{2} \) |
| 11 | \( 1 + 6.24T + 11T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 + 2.64T + 17T^{2} \) |
| 23 | \( 1 + 6.24T + 23T^{2} \) |
| 29 | \( 1 - 3.60T + 29T^{2} \) |
| 31 | \( 1 + 8.24T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + 5.21T + 41T^{2} \) |
| 43 | \( 1 - 4.24T + 43T^{2} \) |
| 47 | \( 1 - 1.60T + 47T^{2} \) |
| 53 | \( 1 + 12.8T + 53T^{2} \) |
| 59 | \( 1 - 1.93T + 59T^{2} \) |
| 61 | \( 1 - 0.0605T + 61T^{2} \) |
| 67 | \( 1 - 6.24T + 67T^{2} \) |
| 71 | \( 1 - 0.329T + 71T^{2} \) |
| 73 | \( 1 + 1.21T + 73T^{2} \) |
| 79 | \( 1 - 0.969T + 79T^{2} \) |
| 83 | \( 1 - 3.67T + 83T^{2} \) |
| 89 | \( 1 - 9.21T + 89T^{2} \) |
| 97 | \( 1 - 9.03T + 97T^{2} \) |
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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
−8.392512404205332452145771197312, −7.964138209583488322206732479562, −7.27375149529527425785430727529, −6.35792213678913352447124944374, −5.66517826480688032382770459141, −4.79143441719933114153996239098, −3.98857568308440947226259945649, −3.22855607241081731270632246762, −2.05514387879784275320835113347, −0.39441577959894301066409407001,
0.39441577959894301066409407001, 2.05514387879784275320835113347, 3.22855607241081731270632246762, 3.98857568308440947226259945649, 4.79143441719933114153996239098, 5.66517826480688032382770459141, 6.35792213678913352447124944374, 7.27375149529527425785430727529, 7.964138209583488322206732479562, 8.392512404205332452145771197312
