| L(s) = 1 | + 64·2-s + 3.07e3·4-s + 1.31e5·8-s + 5.07e4·9-s − 1.44e6·11-s + 5.24e6·16-s + 3.24e6·18-s − 9.24e7·22-s + 3.45e7·23-s + 1.29e7·25-s − 2.59e8·29-s + 2.01e8·32-s + 1.55e8·36-s − 1.27e9·37-s − 2.29e9·43-s − 4.43e9·44-s + 2.21e9·46-s + 8.25e8·50-s − 7.26e9·53-s − 1.65e10·58-s + 7.51e9·64-s − 3.70e9·67-s + 2.57e10·71-s + 6.64e9·72-s − 8.17e10·74-s − 2.54e10·79-s − 2.88e10·81-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 1.41·8-s + 0.286·9-s − 2.70·11-s + 5/4·16-s + 0.404·18-s − 3.82·22-s + 1.12·23-s + 0.264·25-s − 2.34·29-s + 1.06·32-s + 0.429·36-s − 3.02·37-s − 2.38·43-s − 4.05·44-s + 1.58·46-s + 0.373·50-s − 2.38·53-s − 3.31·58-s + 7/8·64-s − 0.335·67-s + 1.69·71-s + 0.404·72-s − 4.28·74-s − 0.928·79-s − 0.918·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s+11/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{13}{2})\) |
|
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 | $C_1$ | \( ( 1 - p^{5} T )^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 626 p^{4} T^{2} + p^{22} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 516088 p^{2} T^{2} + p^{22} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 722404 T + p^{11} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 3314239157192 T^{2} + p^{22} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2526181978784 T^{2} + p^{22} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 159229286268594 T^{2} + p^{22} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 17288500 T + p^{11} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 129579896 T + p^{11} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 42235590219082430 T^{2} + p^{22} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 638597192 T + p^{11} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 21799221948500320 T^{2} + p^{22} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 1147884316 T + p^{11} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 3933313686224804606 T^{2} + p^{22} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 3631326766 T + p^{11} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 5123575775487463810 T^{2} + p^{22} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 28250627620893211272 T^{2} + p^{22} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 1854960384 T + p^{11} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12892526208 T + p^{11} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 78384204329015218272 T^{2} + p^{22} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 12703599400 T + p^{11} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + \)\(58\!\cdots\!22\)\( T^{2} + p^{22} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + \)\(47\!\cdots\!60\)\( T^{2} + p^{22} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + \)\(14\!\cdots\!24\)\( T^{2} + p^{22} T^{4} \) |
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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
−11.30512017186255583749040410804, −11.16152081261108916970228296564, −10.46118063810736483554615567913, −10.20551271242418911411507032410, −9.420127616863764322278222740250, −8.513112493970129175709709665293, −8.011290669596351260733765612927, −7.33212577250122633071355837755, −7.05351929083219659171078518897, −6.24775345805418717330902143457, −5.39511031934079336289677426819, −5.11958392581337934792329660044, −4.88314086997954348557133386619, −3.75724538488783121377340560258, −3.25722374120947985669464034129, −2.76003939019800889511192528738, −1.95691427552733460014189227734, −1.51007823520716416703636463629, 0, 0,
1.51007823520716416703636463629, 1.95691427552733460014189227734, 2.76003939019800889511192528738, 3.25722374120947985669464034129, 3.75724538488783121377340560258, 4.88314086997954348557133386619, 5.11958392581337934792329660044, 5.39511031934079336289677426819, 6.24775345805418717330902143457, 7.05351929083219659171078518897, 7.33212577250122633071355837755, 8.011290669596351260733765612927, 8.513112493970129175709709665293, 9.420127616863764322278222740250, 10.20551271242418911411507032410, 10.46118063810736483554615567913, 11.16152081261108916970228296564, 11.30512017186255583749040410804
