| L(s) = 1 | + 9-s + 8·11-s − 4·13-s + 4·17-s − 6·25-s + 12·29-s + 12·37-s + 8·43-s − 14·49-s − 2·53-s + 8·59-s + 81-s − 12·89-s + 4·97-s + 8·99-s − 24·107-s + 36·113-s − 4·117-s + 26·121-s + 127-s + 131-s + 137-s + 139-s − 32·143-s + 149-s + 151-s + 4·153-s + ⋯ |
| L(s) = 1 | + 1/3·9-s + 2.41·11-s − 1.10·13-s + 0.970·17-s − 6/5·25-s + 2.22·29-s + 1.97·37-s + 1.21·43-s − 2·49-s − 0.274·53-s + 1.04·59-s + 1/9·81-s − 1.27·89-s + 0.406·97-s + 0.804·99-s − 2.32·107-s + 3.38·113-s − 0.369·117-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2.67·143-s + 0.0819·149-s + 0.0813·151-s + 0.323·153-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617984 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617984 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.874615787\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.874615787\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 53 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−8.027167840223797801009792869666, −7.29549770833411599724782132090, −7.09576954771165915285334395134, −6.42897107072744719896577758454, −6.31575357910074254946857805351, −5.83386313895775531748443044492, −5.20742338702140405476830128096, −4.63394453234232935539464379977, −4.25303028692796488061253338187, −3.96538582264402726120940522984, −3.25827357435007527773330566286, −2.78355186146786610328282292507, −2.05606810794439771962311316661, −1.34519509320774605529089606887, −0.809723121056121938776942624986,
0.809723121056121938776942624986, 1.34519509320774605529089606887, 2.05606810794439771962311316661, 2.78355186146786610328282292507, 3.25827357435007527773330566286, 3.96538582264402726120940522984, 4.25303028692796488061253338187, 4.63394453234232935539464379977, 5.20742338702140405476830128096, 5.83386313895775531748443044492, 6.31575357910074254946857805351, 6.42897107072744719896577758454, 7.09576954771165915285334395134, 7.29549770833411599724782132090, 8.027167840223797801009792869666
