| L(s) = 1 | + 8·25-s − 16·29-s + 32·37-s + 10·49-s − 16·53-s − 18·81-s − 16·109-s + 16·113-s + 36·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 40·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | + 8/5·25-s − 2.97·29-s + 5.26·37-s + 10/7·49-s − 2.19·53-s − 2·81-s − 1.53·109-s + 1.50·113-s + 3.27·121-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 + 3.07·169-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 + 0.0708·199-s + 0.0688·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.085290123\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.085290123\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| good | 3 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 5 | $C_2^2$ | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 32 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 + 112 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 68 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 42 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 160 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 170 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−6.40210288891121374094115447771, −6.39788445136666670968901928417, −6.34547423041145641720240943351, −5.78989346589378861338483813858, −5.74267004421504577546789064953, −5.60382945416293231481266440011, −5.56971550302598407697313435535, −5.04719209773789012569326657966, −4.71845585657178657792789896784, −4.64858000794016487762808959184, −4.55769801260544551816222279919, −4.09063132902285450246625602595, −3.95400724380388238606145418735, −3.92062113846217172205235218714, −3.45552803307971704041824596375, −3.03505320270900341416437107200, −2.99638149671264133739516900319, −2.76704869845878984151756616192, −2.47105765567653582470014323750, −2.11316167480912181295180609495, −1.89212013337407305093571910356, −1.41148959404589955066579271609, −1.25527895374484221700690580869, −0.59858846927798077393910490557, −0.52350997919781881894483584735,
0.52350997919781881894483584735, 0.59858846927798077393910490557, 1.25527895374484221700690580869, 1.41148959404589955066579271609, 1.89212013337407305093571910356, 2.11316167480912181295180609495, 2.47105765567653582470014323750, 2.76704869845878984151756616192, 2.99638149671264133739516900319, 3.03505320270900341416437107200, 3.45552803307971704041824596375, 3.92062113846217172205235218714, 3.95400724380388238606145418735, 4.09063132902285450246625602595, 4.55769801260544551816222279919, 4.64858000794016487762808959184, 4.71845585657178657792789896784, 5.04719209773789012569326657966, 5.56971550302598407697313435535, 5.60382945416293231481266440011, 5.74267004421504577546789064953, 5.78989346589378861338483813858, 6.34547423041145641720240943351, 6.39788445136666670968901928417, 6.40210288891121374094115447771
