| L(s) = 1 | + 2·2-s + 3·4-s + 3·7-s + 4·8-s − 5·9-s + 4·11-s + 6·14-s + 5·16-s − 10·18-s + 8·22-s + 8·23-s − 9·25-s + 9·28-s − 10·29-s + 6·32-s − 15·36-s + 16·37-s − 12·43-s + 12·44-s + 16·46-s + 2·49-s − 18·50-s + 18·53-s + 12·56-s − 20·58-s − 15·63-s + 7·64-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 1.13·7-s + 1.41·8-s − 5/3·9-s + 1.20·11-s + 1.60·14-s + 5/4·16-s − 2.35·18-s + 1.70·22-s + 1.66·23-s − 9/5·25-s + 1.70·28-s − 1.85·29-s + 1.06·32-s − 5/2·36-s + 2.63·37-s − 1.82·43-s + 1.80·44-s + 2.35·46-s + 2/7·49-s − 2.54·50-s + 2.47·53-s + 1.60·56-s − 2.62·58-s − 1.88·63-s + 7/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 682276 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 682276 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.617569180\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.617569180\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_2$ | \( 1 - 3 T + p T^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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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
−8.100739732859243393235378198816, −7.930773369176095076170936294898, −7.41975723154921074110230580004, −6.65689015211437814822672486144, −6.58991316168606108179381364137, −5.79238330344911727414565852445, −5.41402537759776928895054430772, −5.39147725414145413474639977518, −4.47276228746620805823938592200, −4.21512786792769606388683659164, −3.51468966459571199068798050891, −3.21495940610495225950895712644, −2.28806177222573667375688542518, −2.00911148220963626129181188337, −0.972858414952871064935415373909,
0.972858414952871064935415373909, 2.00911148220963626129181188337, 2.28806177222573667375688542518, 3.21495940610495225950895712644, 3.51468966459571199068798050891, 4.21512786792769606388683659164, 4.47276228746620805823938592200, 5.39147725414145413474639977518, 5.41402537759776928895054430772, 5.79238330344911727414565852445, 6.58991316168606108179381364137, 6.65689015211437814822672486144, 7.41975723154921074110230580004, 7.930773369176095076170936294898, 8.100739732859243393235378198816
