| L(s) = 1 | − 4·9-s + 8·17-s − 2·25-s − 8·41-s − 12·49-s + 24·73-s − 6·81-s − 40·89-s + 8·97-s + 8·113-s − 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 32·153-s + 157-s + 163-s + 167-s + 44·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | − 4/3·9-s + 1.94·17-s − 2/5·25-s − 1.24·41-s − 1.71·49-s + 2.80·73-s − 2/3·81-s − 4.23·89-s + 0.812·97-s + 0.752·113-s − 1.81·121-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 − 2.58·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3.38·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.066944588\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.066944588\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2}( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 118 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 126 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2}( 1 + 18 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
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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
−6.86137816883771863988117360982, −6.73319427278279329214541701374, −6.60320709852443554873695040843, −6.08849272471504480749388027829, −5.93614132116438903279411387559, −5.91061639709618980634821449642, −5.55378169370215349425242754569, −5.34084132463577479895481249631, −5.21642104113449935245563080241, −4.91337796002406823701912059557, −4.88981382968949787566903416861, −4.22352158976485560969083953699, −4.12360070308573337447723781820, −4.05980074880724542032358770266, −3.49110802942974164431421866743, −3.40064548928288007145308682572, −3.09633989284459406934619731709, −2.86425756751636377550836488887, −2.79979616547071034496730591237, −2.29428942092811512260598712903, −1.84917373363336474364803851102, −1.65305633989897442053963579908, −1.36067334862514455836791140180, −0.71979849040965480667717737248, −0.37258392630966781462872934426,
0.37258392630966781462872934426, 0.71979849040965480667717737248, 1.36067334862514455836791140180, 1.65305633989897442053963579908, 1.84917373363336474364803851102, 2.29428942092811512260598712903, 2.79979616547071034496730591237, 2.86425756751636377550836488887, 3.09633989284459406934619731709, 3.40064548928288007145308682572, 3.49110802942974164431421866743, 4.05980074880724542032358770266, 4.12360070308573337447723781820, 4.22352158976485560969083953699, 4.88981382968949787566903416861, 4.91337796002406823701912059557, 5.21642104113449935245563080241, 5.34084132463577479895481249631, 5.55378169370215349425242754569, 5.91061639709618980634821449642, 5.93614132116438903279411387559, 6.08849272471504480749388027829, 6.60320709852443554873695040843, 6.73319427278279329214541701374, 6.86137816883771863988117360982
