| L(s) = 1 | − 4·2-s + 2·3-s + 12·4-s − 8·6-s − 32·8-s + 9·9-s − 14·11-s + 24·12-s + 80·16-s − 2·17-s − 36·18-s + 34·19-s + 56·22-s − 64·24-s + 46·27-s − 192·32-s − 28·33-s + 8·34-s + 108·36-s − 136·38-s + 46·41-s + 28·43-s − 168·44-s + 160·48-s + 98·49-s − 4·51-s − 184·54-s + ⋯ |
| L(s) = 1 | − 2·2-s + 2/3·3-s + 3·4-s − 4/3·6-s − 4·8-s + 9-s − 1.27·11-s + 2·12-s + 5·16-s − 0.117·17-s − 2·18-s + 1.78·19-s + 2.54·22-s − 8/3·24-s + 1.70·27-s − 6·32-s − 0.848·33-s + 4/17·34-s + 3·36-s − 3.57·38-s + 1.12·41-s + 0.651·43-s − 3.81·44-s + 10/3·48-s + 2·49-s − 0.0784·51-s − 3.40·54-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40000 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.009453294\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.009453294\) |
| \(L(2)\) |
|
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T - 5 T^{2} - 2 p^{2} T^{3} + p^{4} T^{4} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 14 T + 75 T^{2} + 14 p^{2} T^{3} + p^{4} T^{4} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 2 T - 285 T^{2} + 2 p^{2} T^{3} + p^{4} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 34 T + 795 T^{2} - 34 p^{2} T^{3} + p^{4} T^{4} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 46 T + 435 T^{2} - 46 p^{2} T^{3} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 14 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 82 T + p^{2} T^{2} )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 62 T - 645 T^{2} + 62 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T + 14835 T^{2} - 142 p^{2} T^{3} + p^{4} T^{4} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 158 T + 18075 T^{2} + 158 p^{2} T^{3} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 146 T + 13395 T^{2} + 146 p^{2} T^{3} + p^{4} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 94 T + p^{2} T^{2} )^{2} \) |
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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
−12.27883616942764975746933213364, −12.00887304342222067270847484915, −11.04782728594932765000367909273, −10.92184166379660704826162359994, −10.37931986148220533331691829645, −9.841411892061291453868873827766, −9.554508018014441704341197550123, −9.022952541985206914955048899807, −8.532723390981625574022178280883, −7.963742464097824034712416175962, −7.44307623777668857230517091856, −7.35436727818536139680034500717, −6.61879711955856916405278553124, −5.82737591090727636570286472307, −5.29406282903518326366977717440, −4.17703774211329435934034587202, −2.90878777827577758523061980899, −2.85869565789847382378012851866, −1.71579076340713371229261130083, −0.789102809751559382473556484891,
0.789102809751559382473556484891, 1.71579076340713371229261130083, 2.85869565789847382378012851866, 2.90878777827577758523061980899, 4.17703774211329435934034587202, 5.29406282903518326366977717440, 5.82737591090727636570286472307, 6.61879711955856916405278553124, 7.35436727818536139680034500717, 7.44307623777668857230517091856, 7.963742464097824034712416175962, 8.532723390981625574022178280883, 9.022952541985206914955048899807, 9.554508018014441704341197550123, 9.841411892061291453868873827766, 10.37931986148220533331691829645, 10.92184166379660704826162359994, 11.04782728594932765000367909273, 12.00887304342222067270847484915, 12.27883616942764975746933213364
