| L(s) = 1 | + 2-s + 4-s + 2·7-s + 3·8-s − 2·11-s − 5·13-s + 2·14-s + 16-s + 7·17-s + 6·19-s − 2·22-s + 9·23-s − 5·26-s + 2·28-s + 2·29-s − 3·31-s − 32-s + 7·34-s − 37-s + 6·38-s − 3·41-s − 3·43-s − 2·44-s + 9·46-s + 15·47-s − 11·49-s − 5·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.755·7-s + 1.06·8-s − 0.603·11-s − 1.38·13-s + 0.534·14-s + 1/4·16-s + 1.69·17-s + 1.37·19-s − 0.426·22-s + 1.87·23-s − 0.980·26-s + 0.377·28-s + 0.371·29-s − 0.538·31-s − 0.176·32-s + 1.20·34-s − 0.164·37-s + 0.973·38-s − 0.468·41-s − 0.457·43-s − 0.301·44-s + 1.32·46-s + 2.18·47-s − 1.57·49-s − 0.693·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6125625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6125625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.194587105\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.194587105\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - T - p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + 5 T + 28 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 7 T + 42 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 - 9 T + 62 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 3 T + 26 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + T - 32 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 3 T + 46 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 3 T + 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_4$ | \( 1 - 15 T + 146 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 6 T + 98 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 17 T + 186 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 11 T + 148 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + T + 96 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 3 T + 38 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 16 T + 142 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 19 T + 210 T^{2} + 19 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_4$ | \( 1 + 26 T + 330 T^{2} + 26 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 4 T + 181 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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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
−9.009326886671559344557259929405, −8.791921948685192521070249813559, −8.152696016980671943347907021649, −7.896104149932438201226223139528, −7.45999489636388868535554633706, −7.19953411639818651611051226300, −7.03526116404815752596373588659, −6.53010569592905784051965803540, −5.70694313332663637464874323076, −5.37024186115513835272819271057, −5.13059508483270540789814980649, −5.09335446655043069380488745944, −4.22041469467659197613539409522, −4.17117311229390934554463269888, −3.27024468103184251892209335808, −2.95997059778781737590959850581, −2.62542065015625907329286861147, −1.84174331759864344730056862934, −1.38384986319681027676454808451, −0.71026122233452714110055588402,
0.71026122233452714110055588402, 1.38384986319681027676454808451, 1.84174331759864344730056862934, 2.62542065015625907329286861147, 2.95997059778781737590959850581, 3.27024468103184251892209335808, 4.17117311229390934554463269888, 4.22041469467659197613539409522, 5.09335446655043069380488745944, 5.13059508483270540789814980649, 5.37024186115513835272819271057, 5.70694313332663637464874323076, 6.53010569592905784051965803540, 7.03526116404815752596373588659, 7.19953411639818651611051226300, 7.45999489636388868535554633706, 7.896104149932438201226223139528, 8.152696016980671943347907021649, 8.791921948685192521070249813559, 9.009326886671559344557259929405
