| L(s) = 1 | + 3-s − 5-s − 7-s + 9-s + 11-s − 13-s − 15-s − 17-s − 5·19-s − 21-s + 23-s − 4·25-s + 27-s − 2·29-s + 6·31-s + 33-s + 35-s + 8·37-s − 39-s + 5·41-s + 43-s − 45-s + 2·47-s + 49-s − 51-s − 6·53-s − 55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.301·11-s − 0.277·13-s − 0.258·15-s − 0.242·17-s − 1.14·19-s − 0.218·21-s + 0.208·23-s − 4/5·25-s + 0.192·27-s − 0.371·29-s + 1.07·31-s + 0.174·33-s + 0.169·35-s + 1.31·37-s − 0.160·39-s + 0.780·41-s + 0.152·43-s − 0.149·45-s + 0.291·47-s + 1/7·49-s − 0.140·51-s − 0.824·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.902953193\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.902953193\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−8.104318513610742175951051462774, −7.56888740298735314134653487173, −6.66213600242783607941268414532, −6.20751519409090310406187320430, −5.13911723969064542804444169571, −4.22189789896721402110195374631, −3.79823649466850233604500299561, −2.74895958974732954223142492640, −2.06326304576506075758547506793, −0.69687326669233241618134775480,
0.69687326669233241618134775480, 2.06326304576506075758547506793, 2.74895958974732954223142492640, 3.79823649466850233604500299561, 4.22189789896721402110195374631, 5.13911723969064542804444169571, 6.20751519409090310406187320430, 6.66213600242783607941268414532, 7.56888740298735314134653487173, 8.104318513610742175951051462774
