| L(s) = 1 | + 10·5-s + 104·13-s + 132·17-s + 75·25-s − 92·29-s − 48·37-s + 144·41-s − 346·49-s − 124·53-s − 380·61-s + 1.04e3·65-s + 2.15e3·73-s + 1.32e3·85-s − 2.23e3·89-s + 1.66e3·97-s − 540·101-s + 3.49e3·109-s − 1.76e3·113-s + 398·121-s + 500·125-s + 127-s + 131-s + 137-s + 139-s − 920·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 2.21·13-s + 1.88·17-s + 3/5·25-s − 0.589·29-s − 0.213·37-s + 0.548·41-s − 1.00·49-s − 0.321·53-s − 0.797·61-s + 1.98·65-s + 3.45·73-s + 1.68·85-s − 2.65·89-s + 1.74·97-s − 0.532·101-s + 3.06·109-s − 1.46·113-s + 0.299·121-s + 0.357·125-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s − 0.526·145-s + 0.000549·149-s + 0.000538·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.968505831\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.968505831\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 346 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 398 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 p T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 66 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 22974 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 46 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 7058 T^{2} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 24 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 72 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 97486 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 62 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 353298 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 190 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 1766 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 322782 T^{2} + p^{6} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 1078 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 266638 T^{2} + p^{6} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 543814 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 1116 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 834 T + p^{3} T^{2} )^{2} \) |
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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
−9.292408489467636932695143633122, −9.094074067100686678810221388771, −8.434165868490913458985467548450, −8.333862725516490303294986211629, −7.64658497517380748147937196771, −7.59383158208400771342068979637, −6.65282927931518761678220750989, −6.55991780345456703010106489804, −5.98380194938409960625172545185, −5.76319296250611737459743097658, −5.28014519735133870496154064321, −4.95423108890951503159742496235, −4.08652211306604070328189083540, −3.84065923310939226015173632909, −3.11029629006701676661297624103, −3.08980966966120736478202463856, −2.03099962148204467376287099386, −1.64130278522732706370938916534, −1.08737333887223422561925363929, −0.61870669156209468842811600692,
0.61870669156209468842811600692, 1.08737333887223422561925363929, 1.64130278522732706370938916534, 2.03099962148204467376287099386, 3.08980966966120736478202463856, 3.11029629006701676661297624103, 3.84065923310939226015173632909, 4.08652211306604070328189083540, 4.95423108890951503159742496235, 5.28014519735133870496154064321, 5.76319296250611737459743097658, 5.98380194938409960625172545185, 6.55991780345456703010106489804, 6.65282927931518761678220750989, 7.59383158208400771342068979637, 7.64658497517380748147937196771, 8.333862725516490303294986211629, 8.434165868490913458985467548450, 9.094074067100686678810221388771, 9.292408489467636932695143633122
