L(s) = 1 | + 5·3-s + 7·7-s − 2·9-s + 11·11-s − 46·13-s + 127·17-s + 117·19-s + 35·21-s + 80·23-s − 145·27-s + 34·29-s − 292·31-s + 55·33-s − 376·37-s − 230·39-s + 507·41-s + 32·43-s − 134·47-s + 49·49-s + 635·51-s + 612·53-s + 585·57-s + 780·59-s − 426·61-s − 14·63-s − 207·67-s + 400·69-s + ⋯ |
L(s) = 1 | + 0.962·3-s + 0.377·7-s − 0.0740·9-s + 0.301·11-s − 0.981·13-s + 1.81·17-s + 1.41·19-s + 0.363·21-s + 0.725·23-s − 1.03·27-s + 0.217·29-s − 1.69·31-s + 0.290·33-s − 1.67·37-s − 0.944·39-s + 1.93·41-s + 0.113·43-s − 0.415·47-s + 1/7·49-s + 1.74·51-s + 1.58·53-s + 1.35·57-s + 1.72·59-s − 0.894·61-s − 0.0279·63-s − 0.377·67-s + 0.697·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(3.351504385\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.351504385\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - p T \) |
good | 3 | \( 1 - 5 T + p^{3} T^{2} \) |
| 11 | \( 1 - p T + p^{3} T^{2} \) |
| 13 | \( 1 + 46 T + p^{3} T^{2} \) |
| 17 | \( 1 - 127 T + p^{3} T^{2} \) |
| 19 | \( 1 - 117 T + p^{3} T^{2} \) |
| 23 | \( 1 - 80 T + p^{3} T^{2} \) |
| 29 | \( 1 - 34 T + p^{3} T^{2} \) |
| 31 | \( 1 + 292 T + p^{3} T^{2} \) |
| 37 | \( 1 + 376 T + p^{3} T^{2} \) |
| 41 | \( 1 - 507 T + p^{3} T^{2} \) |
| 43 | \( 1 - 32 T + p^{3} T^{2} \) |
| 47 | \( 1 + 134 T + p^{3} T^{2} \) |
| 53 | \( 1 - 612 T + p^{3} T^{2} \) |
| 59 | \( 1 - 780 T + p^{3} T^{2} \) |
| 61 | \( 1 + 426 T + p^{3} T^{2} \) |
| 67 | \( 1 + 207 T + p^{3} T^{2} \) |
| 71 | \( 1 - 702 T + p^{3} T^{2} \) |
| 73 | \( 1 - 1185 T + p^{3} T^{2} \) |
| 79 | \( 1 - 54 T + p^{3} T^{2} \) |
| 83 | \( 1 + 309 T + p^{3} T^{2} \) |
| 89 | \( 1 + 339 T + p^{3} T^{2} \) |
| 97 | \( 1 - 182 T + p^{3} T^{2} \) |
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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
−9.261647182078908842298947724292, −8.382373405481993509497480074273, −7.54394743210727278564005689899, −7.18320087796204247247684281870, −5.62373960323670697667746967895, −5.16038649963684491806500135433, −3.75677558218323073922312354284, −3.11773722187425978147417132367, −2.09501309072694178817612454679, −0.888875161031208308347677340488,
0.888875161031208308347677340488, 2.09501309072694178817612454679, 3.11773722187425978147417132367, 3.75677558218323073922312354284, 5.16038649963684491806500135433, 5.62373960323670697667746967895, 7.18320087796204247247684281870, 7.54394743210727278564005689899, 8.382373405481993509497480074273, 9.261647182078908842298947724292