| L(s) = 1 | − 188·13-s − 50·25-s − 712·37-s + 508·49-s + 2.16e3·61-s + 928·73-s − 4.18e3·97-s + 5.89e3·109-s − 2.67e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.33e4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 4.01·13-s − 2/5·25-s − 3.16·37-s + 1.48·49-s + 4.55·61-s + 1.48·73-s − 4.37·97-s + 5.18·109-s − 2.01·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 6.05·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + 0.000326·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.420994275\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.420994275\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| good | 7 | $C_2^2$ | \( ( 1 - 254 T^{2} + p^{6} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 1339 T^{2} + p^{6} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 47 T + p^{3} T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 - 9385 T^{2} + p^{6} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 9830 T^{2} + p^{6} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 23011 T^{2} + p^{6} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 33649 T^{2} + p^{6} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 58907 T^{2} + p^{6} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 178 T + p^{3} T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 20878 T^{2} + p^{6} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 104339 T^{2} + p^{6} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 113659 T^{2} + p^{6} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 126358 T^{2} + p^{6} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 211066 T^{2} + p^{6} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 542 T + p^{3} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 577226 T^{2} + p^{6} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 10370 T^{2} + p^{6} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 232 T + p^{3} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 864875 T^{2} + p^{6} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 979306 T^{2} + p^{6} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 428798 T^{2} + p^{6} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 1046 T + p^{3} T^{2} )^{4} \) |
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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
−5.99791061500127688941836083757, −5.97381611044847364163438242546, −5.45844759429513828856444396760, −5.24081903078408761523283430502, −5.15295650009407186980454866880, −5.09259096925204621031731919691, −5.05968999715628395222260700937, −4.72945114364608256977946502882, −4.35056165086576368470179048200, −4.13223003103368139136883782474, −3.81583922044414399245772684419, −3.75987979241083916027447460738, −3.63908584709686883285952056317, −3.10462251469429307071810747416, −2.90289645067622904603312060405, −2.59389055792648094388307975367, −2.50049631462422821294937239586, −2.19690992331681416789362765604, −2.08315705161676714305022615568, −1.86302012185652922443439932310, −1.33735204124623489846335512415, −1.17473480166577947928305689713, −0.51967770900283116431401139961, −0.48513258045006321216966010445, −0.18512213008751159904039247649,
0.18512213008751159904039247649, 0.48513258045006321216966010445, 0.51967770900283116431401139961, 1.17473480166577947928305689713, 1.33735204124623489846335512415, 1.86302012185652922443439932310, 2.08315705161676714305022615568, 2.19690992331681416789362765604, 2.50049631462422821294937239586, 2.59389055792648094388307975367, 2.90289645067622904603312060405, 3.10462251469429307071810747416, 3.63908584709686883285952056317, 3.75987979241083916027447460738, 3.81583922044414399245772684419, 4.13223003103368139136883782474, 4.35056165086576368470179048200, 4.72945114364608256977946502882, 5.05968999715628395222260700937, 5.09259096925204621031731919691, 5.15295650009407186980454866880, 5.24081903078408761523283430502, 5.45844759429513828856444396760, 5.97381611044847364163438242546, 5.99791061500127688941836083757
