L(s) = 1 | − 3-s + 5-s + 3·7-s + 9-s + 5·11-s + 13-s − 15-s + 5·17-s − 2·19-s − 3·21-s + 23-s + 25-s − 27-s + 10·29-s + 2·31-s − 5·33-s + 3·35-s − 3·37-s − 39-s − 9·41-s + 4·43-s + 45-s − 10·47-s + 2·49-s − 5·51-s + 9·53-s + 5·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1.13·7-s + 1/3·9-s + 1.50·11-s + 0.277·13-s − 0.258·15-s + 1.21·17-s − 0.458·19-s − 0.654·21-s + 0.208·23-s + 1/5·25-s − 0.192·27-s + 1.85·29-s + 0.359·31-s − 0.870·33-s + 0.507·35-s − 0.493·37-s − 0.160·39-s − 1.40·41-s + 0.609·43-s + 0.149·45-s − 1.45·47-s + 2/7·49-s − 0.700·51-s + 1.23·53-s + 0.674·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.347076694\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.347076694\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 11 T + p T^{2} \) |
| 97 | \( 1 + 9 T + p T^{2} \) |
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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
−8.582849055534389147053429823048, −8.092828186253584702384612147549, −6.98970109409746550580962933403, −6.45143952249917777900921807109, −5.64070594873197162301522629244, −4.86218754716371112791943230568, −4.18115076091625241953136432080, −3.11202367766983880548407088633, −1.69680021454953043072253508965, −1.09737019560828316914116411893,
1.09737019560828316914116411893, 1.69680021454953043072253508965, 3.11202367766983880548407088633, 4.18115076091625241953136432080, 4.86218754716371112791943230568, 5.64070594873197162301522629244, 6.45143952249917777900921807109, 6.98970109409746550580962933403, 8.092828186253584702384612147549, 8.582849055534389147053429823048