| L(s) = 1 | − 3·9-s + 4·13-s + 10·25-s − 20·37-s − 28·61-s − 20·73-s + 9·81-s + 28·97-s − 4·109-s − 12·117-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 14·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | − 9-s + 1.10·13-s + 2·25-s − 3.28·37-s − 3.58·61-s − 2.34·73-s + 81-s + 2.84·97-s − 0.383·109-s − 1.10·117-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.07·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.459461633\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.459461633\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
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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
−9.216816332137165395494573315500, −8.763354738188187608292141215526, −8.685604204031375135554107417137, −7.992907980026055313790914412330, −7.76077986013522224737288385081, −7.16536976650056772683255048129, −6.89421756484277699151705267606, −6.40303506450071400686339200771, −6.09547455772947283310479794247, −5.69174789907543695266489746142, −5.23641759229527618688331756827, −4.77666763692699261946312198812, −4.56068834799076308527594398306, −3.60181559733772757475246217339, −3.56668456363303578500709839870, −2.93526184693426707344651788613, −2.64928102845260039095935617264, −1.65679254041983229888003939170, −1.43986159162800746822821084039, −0.41531380651512062279424756198,
0.41531380651512062279424756198, 1.43986159162800746822821084039, 1.65679254041983229888003939170, 2.64928102845260039095935617264, 2.93526184693426707344651788613, 3.56668456363303578500709839870, 3.60181559733772757475246217339, 4.56068834799076308527594398306, 4.77666763692699261946312198812, 5.23641759229527618688331756827, 5.69174789907543695266489746142, 6.09547455772947283310479794247, 6.40303506450071400686339200771, 6.89421756484277699151705267606, 7.16536976650056772683255048129, 7.76077986013522224737288385081, 7.992907980026055313790914412330, 8.685604204031375135554107417137, 8.763354738188187608292141215526, 9.216816332137165395494573315500
