| L(s) = 1 | + 3·4-s − 18·9-s − 22·11-s − 7·16-s + 36·31-s − 54·36-s − 66·44-s + 78·49-s + 204·59-s − 69·64-s − 156·71-s + 243·81-s − 4·89-s + 396·99-s + 363·121-s + 108·124-s + 127-s + 131-s + 137-s + 139-s + 126·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 162·169-s + ⋯ |
| L(s) = 1 | + 3/4·4-s − 2·9-s − 2·11-s − 0.437·16-s + 1.16·31-s − 3/2·36-s − 3/2·44-s + 1.59·49-s + 3.45·59-s − 1.07·64-s − 2.19·71-s + 3·81-s − 0.0449·89-s + 4·99-s + 3·121-s + 0.870·124-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 7/8·144-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 0.958·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75625 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.178953998\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.178953998\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{4} T^{4} \) |
| 3 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 78 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 162 T^{2} + p^{4} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 402 T^{2} + p^{4} T^{4} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 18 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 3522 T^{2} + p^{4} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 102 T + p^{2} T^{2} )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 78 T + p^{2} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 10638 T^{2} + p^{4} T^{4} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 13602 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
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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
−12.07020504303964193814350057999, −11.29726050856000625584511821149, −11.14217281421102423719063014103, −10.40732354593259595893622766825, −10.32619649196324180990290323690, −9.585303737178747379373860245611, −8.836679816951301496318980046595, −8.453233855848391218900478210461, −8.201199412293018912980845657426, −7.42905760215809890801081127300, −7.15132191124381780732299313182, −6.28704995236492052386551409764, −5.90779443231534368574359283289, −5.34295149301340173925581710551, −4.97348069489792911722369889304, −4.02017827111228033144535450867, −3.02015473333888639594364688398, −2.65084380526423830416600307595, −2.20617296935051957870374440940, −0.50866287705866535203923809903,
0.50866287705866535203923809903, 2.20617296935051957870374440940, 2.65084380526423830416600307595, 3.02015473333888639594364688398, 4.02017827111228033144535450867, 4.97348069489792911722369889304, 5.34295149301340173925581710551, 5.90779443231534368574359283289, 6.28704995236492052386551409764, 7.15132191124381780732299313182, 7.42905760215809890801081127300, 8.201199412293018912980845657426, 8.453233855848391218900478210461, 8.836679816951301496318980046595, 9.585303737178747379373860245611, 10.32619649196324180990290323690, 10.40732354593259595893622766825, 11.14217281421102423719063014103, 11.29726050856000625584511821149, 12.07020504303964193814350057999
