| L(s) = 1 | − 4-s + 2·9-s + 16-s − 2·36-s − 49-s − 64-s − 4·71-s + 4·79-s + 3·81-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 2·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 196-s + ⋯ |
| L(s) = 1 | − 4-s + 2·9-s + 16-s − 2·36-s − 49-s − 64-s − 4·71-s + 4·79-s + 3·81-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 2·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 196-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.024322428\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.024322428\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 71 | $C_1$ | \( ( 1 + T )^{4} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 79 | $C_1$ | \( ( 1 - T )^{4} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
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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
−9.765012748437688413368988412335, −9.602136215530039227366546270583, −9.273086296306183427017120347383, −8.866905105064671329983701953689, −8.278973365732278437695112623326, −8.002590247929937488962729954525, −7.55305362595740159195052512078, −7.16429588126950139333433034383, −6.81684790072022592688546797161, −6.23411258645905093824127678128, −5.87814879234559160367791851818, −5.22797957112767086184052552108, −4.83873280622959609951008350304, −4.32135749502739859126499642656, −4.27687517674649775287371903480, −3.42991126237455350429414279205, −3.24988279972240346684076149846, −2.19301970758838871135275432597, −1.62674609263186377695371728245, −0.954165853816513525998763122907,
0.954165853816513525998763122907, 1.62674609263186377695371728245, 2.19301970758838871135275432597, 3.24988279972240346684076149846, 3.42991126237455350429414279205, 4.27687517674649775287371903480, 4.32135749502739859126499642656, 4.83873280622959609951008350304, 5.22797957112767086184052552108, 5.87814879234559160367791851818, 6.23411258645905093824127678128, 6.81684790072022592688546797161, 7.16429588126950139333433034383, 7.55305362595740159195052512078, 8.002590247929937488962729954525, 8.278973365732278437695112623326, 8.866905105064671329983701953689, 9.273086296306183427017120347383, 9.602136215530039227366546270583, 9.765012748437688413368988412335
