L(s) = 1 | − 9-s + 4·17-s − 25-s − 20·41-s + 16·47-s − 14·49-s + 16·71-s − 20·73-s + 81-s + 12·89-s + 4·97-s + 32·103-s + 4·113-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 4·153-s + 157-s + 163-s + 167-s + 22·169-s + 173-s + 179-s + ⋯ |
L(s) = 1 | − 1/3·9-s + 0.970·17-s − 1/5·25-s − 3.12·41-s + 2.33·47-s − 2·49-s + 1.89·71-s − 2.34·73-s + 1/9·81-s + 1.27·89-s + 0.406·97-s + 3.15·103-s + 0.376·113-s + 6/11·121-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 − 0.323·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.961688882\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.961688882\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
good | 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−8.551463842332336419378041239689, −8.412833569384716939795128549500, −7.84901833642044963591966113597, −7.69042905905780937935050475279, −7.20355571090021147374732360017, −6.86524450594579777159424397377, −6.41097366005321248085380167058, −6.15173484194683459295153413130, −5.59186205326040074770112823850, −5.39289641792661234315080691248, −4.89790522643415220046098074564, −4.61497332374580121720204879283, −4.00813137038554161732793385360, −3.59684878330961196726182090427, −3.10020302982126717907513491282, −3.00649717045108822405187959438, −1.96457036027455283017002482678, −1.96150747691952472582251646366, −1.10871868552731782992506529266, −0.44274378753751550386415797370,
0.44274378753751550386415797370, 1.10871868552731782992506529266, 1.96150747691952472582251646366, 1.96457036027455283017002482678, 3.00649717045108822405187959438, 3.10020302982126717907513491282, 3.59684878330961196726182090427, 4.00813137038554161732793385360, 4.61497332374580121720204879283, 4.89790522643415220046098074564, 5.39289641792661234315080691248, 5.59186205326040074770112823850, 6.15173484194683459295153413130, 6.41097366005321248085380167058, 6.86524450594579777159424397377, 7.20355571090021147374732360017, 7.69042905905780937935050475279, 7.84901833642044963591966113597, 8.412833569384716939795128549500, 8.551463842332336419378041239689