| L(s) = 1 | − 4-s − 2·5-s − 10·11-s + 16-s + 2·20-s − 25-s − 6·29-s + 2·31-s + 2·41-s + 10·44-s + 13·49-s + 20·55-s − 16·59-s + 10·61-s − 64-s + 12·71-s − 2·80-s + 16·89-s + 100-s − 20·101-s − 2·109-s + 6·116-s + 53·121-s − 2·124-s + 12·125-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 0.894·5-s − 3.01·11-s + 1/4·16-s + 0.447·20-s − 1/5·25-s − 1.11·29-s + 0.359·31-s + 0.312·41-s + 1.50·44-s + 13/7·49-s + 2.69·55-s − 2.08·59-s + 1.28·61-s − 1/8·64-s + 1.42·71-s − 0.223·80-s + 1.69·89-s + 1/10·100-s − 1.99·101-s − 0.191·109-s + 0.557·116-s + 4.81·121-s − 0.179·124-s + 1.07·125-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11088900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11088900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3979652671\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3979652671\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 37 | $C_2$ | \( 1 + T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 33 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 97 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 193 T^{2} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.612018417221854147724337855600, −8.335161604938855825475387642571, −8.050300285049835727730483842023, −7.68980325657955844802350644398, −7.56528430257447074955363167617, −7.10501134937830932904269060839, −6.70914310679236794270745960890, −5.95775912192599789269712483037, −5.73778400578310807801356618597, −5.32498614529446413303903557158, −5.09529117194524344805585489636, −4.44446803762514620469864485516, −4.38245273044233622472990910587, −3.58905394884931860459991501782, −3.40363180219163034709022624486, −2.76434728669581474791762038211, −2.37996817428331787590801346737, −1.95636956043618698503986375288, −0.901878546130368232964868803612, −0.23854590838559594356905553229,
0.23854590838559594356905553229, 0.901878546130368232964868803612, 1.95636956043618698503986375288, 2.37996817428331787590801346737, 2.76434728669581474791762038211, 3.40363180219163034709022624486, 3.58905394884931860459991501782, 4.38245273044233622472990910587, 4.44446803762514620469864485516, 5.09529117194524344805585489636, 5.32498614529446413303903557158, 5.73778400578310807801356618597, 5.95775912192599789269712483037, 6.70914310679236794270745960890, 7.10501134937830932904269060839, 7.56528430257447074955363167617, 7.68980325657955844802350644398, 8.050300285049835727730483842023, 8.335161604938855825475387642571, 8.612018417221854147724337855600
