L(s) = 1 | + 4-s − 5-s − 4·9-s + 3·11-s + 16-s − 9·19-s − 20-s − 4·25-s + 10·29-s + 31-s − 4·36-s − 7·41-s + 3·44-s + 4·45-s + 49-s − 3·55-s + 8·59-s − 2·61-s + 64-s + 25·71-s − 9·76-s − 79-s − 80-s + 7·81-s − 9·89-s + 9·95-s − 12·99-s + ⋯ |
L(s) = 1 | + 1/2·4-s − 0.447·5-s − 4/3·9-s + 0.904·11-s + 1/4·16-s − 2.06·19-s − 0.223·20-s − 4/5·25-s + 1.85·29-s + 0.179·31-s − 2/3·36-s − 1.09·41-s + 0.452·44-s + 0.596·45-s + 1/7·49-s − 0.404·55-s + 1.04·59-s − 0.256·61-s + 1/8·64-s + 2.96·71-s − 1.03·76-s − 0.112·79-s − 0.111·80-s + 7/9·81-s − 0.953·89-s + 0.923·95-s − 1.20·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7115995980\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7115995980\) |
\(L(\frac{3}{2})\) |
|
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_2$ | \( 1 + T + p T^{2} \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 9 T + p T^{2} ) \) |
good | 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 9 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 40 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 73 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 9 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 60 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 131 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
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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
−12.62612934929032554678336338024, −12.22336667625757416270395253552, −11.60381783260442565766016521761, −11.22633663941716090666286842996, −10.55578264536791298876246669639, −9.910364646883300315518125241244, −8.976851365462203903291224173433, −8.375065750562238886292296086356, −8.066384233774645269096311711156, −6.77891256781486307782173841295, −6.48142065792258831297629434451, −5.62780188754507505773141895568, −4.51474613651913205199474225191, −3.57421905812722368910314872805, −2.37516654802028633890714464243,
2.37516654802028633890714464243, 3.57421905812722368910314872805, 4.51474613651913205199474225191, 5.62780188754507505773141895568, 6.48142065792258831297629434451, 6.77891256781486307782173841295, 8.066384233774645269096311711156, 8.375065750562238886292296086356, 8.976851365462203903291224173433, 9.910364646883300315518125241244, 10.55578264536791298876246669639, 11.22633663941716090666286842996, 11.60381783260442565766016521761, 12.22336667625757416270395253552, 12.62612934929032554678336338024