L(s) = 1 | + 2-s + 4-s + 4·7-s + 8-s − 4·11-s + 4·14-s + 16-s + 2·17-s + 19-s − 4·22-s + 2·23-s − 5·25-s + 4·28-s + 6·29-s + 6·31-s + 32-s + 2·34-s − 8·37-s + 38-s − 10·41-s − 12·43-s − 4·44-s + 2·46-s − 10·47-s + 9·49-s − 5·50-s − 2·53-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.51·7-s + 0.353·8-s − 1.20·11-s + 1.06·14-s + 1/4·16-s + 0.485·17-s + 0.229·19-s − 0.852·22-s + 0.417·23-s − 25-s + 0.755·28-s + 1.11·29-s + 1.07·31-s + 0.176·32-s + 0.342·34-s − 1.31·37-s + 0.162·38-s − 1.56·41-s − 1.82·43-s − 0.603·44-s + 0.294·46-s − 1.45·47-s + 9/7·49-s − 0.707·50-s − 0.274·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.170527203\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.170527203\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−11.66713901533468866139416925398, −10.75340342595167971109549852009, −9.973211833044553721935251130017, −8.306705427352619616134264570074, −7.88755447490832193535409441723, −6.62468423904919083533508450719, −5.21883010312481367678358660757, −4.81736217668307600490535491911, −3.23599181183020074918802290096, −1.79376692971899737144429675124,
1.79376692971899737144429675124, 3.23599181183020074918802290096, 4.81736217668307600490535491911, 5.21883010312481367678358660757, 6.62468423904919083533508450719, 7.88755447490832193535409441723, 8.306705427352619616134264570074, 9.973211833044553721935251130017, 10.75340342595167971109549852009, 11.66713901533468866139416925398