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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6532992219\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6532992219\) |
\(L(1)\) |
|
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
−11.86043896787044078514714658870, −10.93922014141176463092218787469, −10.55112336478754415447079823366, −9.123962078617479028552404123501, −8.118040181241947617929690191827, −6.99886877428203207051998883876, −5.95821846262433145702348962949, −4.99401846089250571451850638538, −3.90660080233406067502921605051, −1.68424814305526199645363431078,
1.68424814305526199645363431078, 3.90660080233406067502921605051, 4.99401846089250571451850638538, 5.95821846262433145702348962949, 6.99886877428203207051998883876, 8.118040181241947617929690191827, 9.123962078617479028552404123501, 10.55112336478754415447079823366, 10.93922014141176463092218787469, 11.86043896787044078514714658870