| L(s) = 1 | − 12·11-s − 130·19-s + 132·29-s + 598·31-s − 720·41-s + 517·49-s + 1.57e3·59-s + 934·61-s + 720·71-s − 544·79-s + 3.62e3·101-s + 3.08e3·109-s − 2.55e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4.36e3·169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | − 0.328·11-s − 1.56·19-s + 0.845·29-s + 3.46·31-s − 2.74·41-s + 1.50·49-s + 3.46·59-s + 1.96·61-s + 1.20·71-s − 0.774·79-s + 3.57·101-s + 2.71·109-s − 1.91·121-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 + 1.98·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.248014644\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.248014644\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 517 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 6 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 4369 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 3742 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 65 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 p^{2} T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 66 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 299 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 396 T + p^{3} T^{2} )( 1 + 396 T + p^{3} T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 360 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 117805 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 201562 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 38 p^{2} T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 786 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 467 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 554437 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 360 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 696238 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 272 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 p^{2} T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1564225 T^{2} + p^{6} T^{4} \) |
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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
−10.04896351344906522153930293156, −9.771042573648572202317728756913, −8.833270990939616106694079281752, −8.614096959604641905132783406138, −8.274597258277529531505261262341, −8.137591235734430110119987364919, −7.23927612463231089013330079382, −6.93539286169458908288961638285, −6.38914694397352921214603583239, −6.32460255105483060678170286882, −5.43885620234813102745209619323, −5.17392451792886768135955983190, −4.44904969566377452666790772657, −4.31414653190913531999558808223, −3.52664206174124560308414985471, −3.03528590723630245259635650061, −2.19449942399162125691817537076, −2.16307114970417391183623631108, −0.885675107400046330724333032509, −0.60419905632165308655559250623,
0.60419905632165308655559250623, 0.885675107400046330724333032509, 2.16307114970417391183623631108, 2.19449942399162125691817537076, 3.03528590723630245259635650061, 3.52664206174124560308414985471, 4.31414653190913531999558808223, 4.44904969566377452666790772657, 5.17392451792886768135955983190, 5.43885620234813102745209619323, 6.32460255105483060678170286882, 6.38914694397352921214603583239, 6.93539286169458908288961638285, 7.23927612463231089013330079382, 8.137591235734430110119987364919, 8.274597258277529531505261262341, 8.614096959604641905132783406138, 8.833270990939616106694079281752, 9.771042573648572202317728756913, 10.04896351344906522153930293156
