L(s) = 1 | + 3-s + 5-s + 9-s + 4·11-s − 2·13-s + 15-s + 2·17-s − 4·19-s + 25-s + 27-s − 2·29-s + 4·33-s − 10·37-s − 2·39-s + 10·41-s − 4·43-s + 45-s − 8·47-s − 7·49-s + 2·51-s − 10·53-s + 4·55-s − 4·57-s + 4·59-s − 2·61-s − 2·65-s − 12·67-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1/3·9-s + 1.20·11-s − 0.554·13-s + 0.258·15-s + 0.485·17-s − 0.917·19-s + 1/5·25-s + 0.192·27-s − 0.371·29-s + 0.696·33-s − 1.64·37-s − 0.320·39-s + 1.56·41-s − 0.609·43-s + 0.149·45-s − 1.16·47-s − 49-s + 0.280·51-s − 1.37·53-s + 0.539·55-s − 0.529·57-s + 0.520·59-s − 0.256·61-s − 0.248·65-s − 1.46·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.596242222\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.596242222\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.26787268519799195488690762647, −11.16070118695890574879383709815, −10.00510556501914029619044642425, −9.265565204604989643425731491439, −8.330811446905088069540074876671, −7.12360478295630625617848815878, −6.13587545605028508104380795249, −4.68831052899409581492283307982, −3.39274372810076015084631955767, −1.82346513010869405208007428430,
1.82346513010869405208007428430, 3.39274372810076015084631955767, 4.68831052899409581492283307982, 6.13587545605028508104380795249, 7.12360478295630625617848815878, 8.330811446905088069540074876671, 9.265565204604989643425731491439, 10.00510556501914029619044642425, 11.16070118695890574879383709815, 12.26787268519799195488690762647