| L(s) = 1 | − 2·3-s + 3·9-s + 4·11-s + 12·17-s − 2·19-s − 6·25-s − 4·27-s − 8·33-s − 16·41-s − 16·43-s − 14·49-s − 24·51-s + 4·57-s + 24·67-s + 12·73-s + 12·75-s + 5·81-s + 12·83-s − 4·97-s + 12·99-s + 16·107-s − 32·113-s − 10·121-s + 32·123-s + 127-s + 32·129-s + 131-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 9-s + 1.20·11-s + 2.91·17-s − 0.458·19-s − 6/5·25-s − 0.769·27-s − 1.39·33-s − 2.49·41-s − 2.43·43-s − 2·49-s − 3.36·51-s + 0.529·57-s + 2.93·67-s + 1.40·73-s + 1.38·75-s + 5/9·81-s + 1.31·83-s − 0.406·97-s + 1.20·99-s + 1.54·107-s − 3.01·113-s − 0.909·121-s + 2.88·123-s + 0.0887·127-s + 2.81·129-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 831744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 831744 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.351834232\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.351834232\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 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
−8.141189095841594822687515621678, −8.010861038875055673500739752797, −6.98540924976438206924302865152, −6.95937341559717252012161118029, −6.41430831950023271755720232491, −6.00528925290387781663305320049, −5.46352487964834549332775040541, −5.11512058941944572023463227823, −4.77229699453803063623408370641, −3.88435800467627005044970379170, −3.51636850061813590584504045751, −3.23825513147504544346615507462, −1.85229865463954971789789038207, −1.55947103904026918949790889898, −0.63216328103521466765374947899,
0.63216328103521466765374947899, 1.55947103904026918949790889898, 1.85229865463954971789789038207, 3.23825513147504544346615507462, 3.51636850061813590584504045751, 3.88435800467627005044970379170, 4.77229699453803063623408370641, 5.11512058941944572023463227823, 5.46352487964834549332775040541, 6.00528925290387781663305320049, 6.41430831950023271755720232491, 6.95937341559717252012161118029, 6.98540924976438206924302865152, 8.010861038875055673500739752797, 8.141189095841594822687515621678
