L(s) = 1 | + 5-s + 5·9-s + 3·11-s − 4·16-s − 4·25-s − 10·29-s − 31-s + 3·41-s + 5·45-s + 5·49-s + 3·55-s + 5·59-s − 2·61-s + 71-s − 5·79-s − 4·80-s + 16·81-s + 10·89-s + 15·99-s − 4·101-s − 10·109-s − 9·121-s − 9·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 5/3·9-s + 0.904·11-s − 16-s − 4/5·25-s − 1.85·29-s − 0.179·31-s + 0.468·41-s + 0.745·45-s + 5/7·49-s + 0.404·55-s + 0.650·59-s − 0.256·61-s + 0.118·71-s − 0.562·79-s − 0.447·80-s + 16/9·81-s + 1.05·89-s + 1.50·99-s − 0.398·101-s − 0.957·109-s − 0.818·121-s − 0.804·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.428053298\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.428053298\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 5 | $C_2$ | \( 1 - T + p T^{2} \) |
| 31 | $C_2$ | \( 1 + T + p T^{2} \) |
good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 15 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 15 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 55 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 95 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 95 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
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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
−10.70917705909503306583588671539, −10.05458370108966505514733389242, −9.621828832141109284258005784986, −9.239371053510812335337630973328, −8.776950382982104263607468888499, −7.83605235154742652087022721420, −7.34589008364915840446400319821, −6.89542555928777320395612330756, −6.29973140469120430294973959024, −5.64025634937085876476102581393, −4.84189736523501903395160367137, −4.09045033490687232695717305489, −3.73687090201400686590050399962, −2.29562056225529276258388423470, −1.51635649866021460384122379637,
1.51635649866021460384122379637, 2.29562056225529276258388423470, 3.73687090201400686590050399962, 4.09045033490687232695717305489, 4.84189736523501903395160367137, 5.64025634937085876476102581393, 6.29973140469120430294973959024, 6.89542555928777320395612330756, 7.34589008364915840446400319821, 7.83605235154742652087022721420, 8.776950382982104263607468888499, 9.239371053510812335337630973328, 9.621828832141109284258005784986, 10.05458370108966505514733389242, 10.70917705909503306583588671539