| L(s) = 1 | + 1.34e3·25-s − 9.60e3·49-s + 4.22e4·73-s − 7.48e4·97-s − 5.85e4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 956·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
| L(s) = 1 | + 2.15·25-s − 4·49-s + 7.92·73-s − 7.95·97-s − 4·121-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s + 4.50e−5·149-s + 4.38e−5·151-s + 4.05e−5·157-s + 3.76e−5·163-s + 3.58e−5·167-s + 0.0334·169-s + 3.34e−5·173-s + 3.12e−5·179-s + 3.05e−5·181-s + 2.74e−5·191-s + 2.68e−5·193-s + 2.57e−5·197-s + 2.52e−5·199-s + 2.24e−5·211-s + 2.01e−5·223-s + 1.94e−5·227-s + 1.90e−5·229-s + 1.84e−5·233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.7882533224\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7882533224\) |
| \(L(3)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 - 672 T^{2} + p^{8} T^{4} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{4} \) |
| 11 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 - 240 T + p^{4} T^{2} )^{2}( 1 + 240 T + p^{4} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 154560 T^{2} + p^{8} T^{4} )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{4}( 1 + p^{2} T )^{4} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{4}( 1 + p^{2} T )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 + 137760 T^{2} + p^{8} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2162 T + p^{4} T^{2} )^{2}( 1 + 2162 T + p^{4} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 4374720 T^{2} + p^{8} T^{4} )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{4}( 1 + p^{2} T )^{4} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{4}( 1 + p^{2} T )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 - 12509280 T^{2} + p^{8} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 - 6958 T + p^{4} T^{2} )^{2}( 1 + 6958 T + p^{4} T^{2} )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{4}( 1 + p^{2} T )^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{4}( 1 + p^{2} T )^{4} \) |
| 73 | $C_2$ | \( ( 1 - 10560 T + p^{4} T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{4} \) |
| 83 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{4} \) |
| 89 | $C_2^2$ | \( ( 1 + 121779840 T^{2} + p^{8} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 18720 T + p^{4} T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.47092876856752099201233419165, −6.33561218054273820426399519917, −6.29464238575916581913667246602, −5.56488050194677280796988217088, −5.31038513283702210577144103362, −5.28401953386302939380255766921, −5.16012271771329061693568973682, −5.09732222260479878408160935313, −4.44959700134859732826294515962, −4.40117607063412088941936224306, −4.30216956220706161239601758286, −3.69473416248439531812493915410, −3.57083085517741561393896632546, −3.55958149660570249200866395435, −3.10363804760620292875707558210, −2.76109493493058694139961458588, −2.54318766127922846877473978730, −2.52457870166914904805372426669, −1.97131160794564611997223165310, −1.69164685527601505982600356270, −1.40848708688443499730356699667, −1.04141066028886556959907427846, −0.990332650397682746876906820316, −0.45335212681523934586054292632, −0.10237640529779271402195796737,
0.10237640529779271402195796737, 0.45335212681523934586054292632, 0.990332650397682746876906820316, 1.04141066028886556959907427846, 1.40848708688443499730356699667, 1.69164685527601505982600356270, 1.97131160794564611997223165310, 2.52457870166914904805372426669, 2.54318766127922846877473978730, 2.76109493493058694139961458588, 3.10363804760620292875707558210, 3.55958149660570249200866395435, 3.57083085517741561393896632546, 3.69473416248439531812493915410, 4.30216956220706161239601758286, 4.40117607063412088941936224306, 4.44959700134859732826294515962, 5.09732222260479878408160935313, 5.16012271771329061693568973682, 5.28401953386302939380255766921, 5.31038513283702210577144103362, 5.56488050194677280796988217088, 6.29464238575916581913667246602, 6.33561218054273820426399519917, 6.47092876856752099201233419165
