| L(s) = 1 | + 48·19-s + 152·31-s − 104·49-s − 280·61-s + 120·79-s − 296·109-s + 288·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 152·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯ |
| L(s) = 1 | + 2.52·19-s + 4.90·31-s − 2.12·49-s − 4.59·61-s + 1.51·79-s − 2.71·109-s + 2.38·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 0.899·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + 0.00448·223-s + 0.00440·227-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(9.047891706\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.047891706\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( ( 1 + 52 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 144 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 76 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 210 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 12 T + p^{2} T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 - 42 p T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 1610 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 38 T + p^{2} T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + 2692 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 1440 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 902 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 1470 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 + p^{4} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 70 T + p^{2} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 2798 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 4030 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 10474 T^{2} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 30 T + p^{2} T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 4254 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 14784 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 9802 T^{2} + p^{4} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.05120399470625085635995113217, −5.64286915212832623973115177342, −5.28985344038386501694150649259, −5.22432894846562201183827789812, −5.20950698049703792464523122467, −4.83760966741955014028638208182, −4.52082221631845144632464170203, −4.49995855439480514726950041527, −4.48577917544116495308703653563, −4.06642463558453882537728383312, −3.82457764040428256666741174196, −3.51836289982596967452324126362, −3.31214836313024975437119514023, −2.94057182174775817946711839736, −2.93340601434532563561087666797, −2.80690102766861581351560880229, −2.75903552614264706230778618454, −2.21581544820964905479859790059, −1.76123179039136451161783926023, −1.63156708963366023571402211044, −1.47155557383070521239838784766, −1.12978006936586634143363347982, −0.70041212499084396180832474200, −0.65733936061242947546149205661, −0.33985099148076055497670750003,
0.33985099148076055497670750003, 0.65733936061242947546149205661, 0.70041212499084396180832474200, 1.12978006936586634143363347982, 1.47155557383070521239838784766, 1.63156708963366023571402211044, 1.76123179039136451161783926023, 2.21581544820964905479859790059, 2.75903552614264706230778618454, 2.80690102766861581351560880229, 2.93340601434532563561087666797, 2.94057182174775817946711839736, 3.31214836313024975437119514023, 3.51836289982596967452324126362, 3.82457764040428256666741174196, 4.06642463558453882537728383312, 4.48577917544116495308703653563, 4.49995855439480514726950041527, 4.52082221631845144632464170203, 4.83760966741955014028638208182, 5.20950698049703792464523122467, 5.22432894846562201183827789812, 5.28985344038386501694150649259, 5.64286915212832623973115177342, 6.05120399470625085635995113217
