L(s) = 1 | + 3-s − 7-s + 9-s + 13-s + 19-s − 21-s + 27-s + 31-s − 2·37-s + 39-s − 43-s + 57-s − 61-s − 63-s − 67-s − 2·73-s − 2·79-s + 81-s − 91-s + 93-s + 97-s + 2·103-s − 109-s − 2·111-s + 117-s + ⋯ |
L(s) = 1 | + 3-s − 7-s + 9-s + 13-s + 19-s − 21-s + 27-s + 31-s − 2·37-s + 39-s − 43-s + 57-s − 61-s − 63-s − 67-s − 2·73-s − 2·79-s + 81-s − 91-s + 93-s + 97-s + 2·103-s − 109-s − 2·111-s + 117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.418831203\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.418831203\) |
\(L(1)\) |
|
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 | \( 1 \) |
good | 7 | \( 1 + T + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( 1 - T + T^{2} \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( 1 - T + T^{2} \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( 1 - T + T^{2} \) |
| 37 | \( ( 1 + T )^{2} \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( 1 + T + T^{2} \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( 1 + T + T^{2} \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 + T )^{2} \) |
| 79 | \( ( 1 + T )^{2} \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( 1 - T + T^{2} \) |
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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
−9.942358794411713334637317422438, −9.008379091962069376289140371872, −8.514887001081074491570988415409, −7.49708229508296610459485823383, −6.76021732868365418569048814287, −5.87871841802009260578597704877, −4.62591002314671237001825315526, −3.49591326774562990465276500172, −3.00932761946599191564241934794, −1.53259497709663034196923246681,
1.53259497709663034196923246681, 3.00932761946599191564241934794, 3.49591326774562990465276500172, 4.62591002314671237001825315526, 5.87871841802009260578597704877, 6.76021732868365418569048814287, 7.49708229508296610459485823383, 8.514887001081074491570988415409, 9.008379091962069376289140371872, 9.942358794411713334637317422438