L(s) = 1 | + 4-s + 5-s + 7-s − 11-s − 13-s + 16-s + 17-s + 20-s − 2·23-s + 25-s + 28-s + 35-s − 2·37-s − 41-s − 44-s − 52-s + 53-s − 55-s − 59-s − 61-s + 64-s − 65-s + 67-s + 68-s − 71-s + 73-s − 77-s + ⋯ |
L(s) = 1 | + 4-s + 5-s + 7-s − 11-s − 13-s + 16-s + 17-s + 20-s − 2·23-s + 25-s + 28-s + 35-s − 2·37-s − 41-s − 44-s − 52-s + 53-s − 55-s − 59-s − 61-s + 64-s − 65-s + 67-s + 68-s − 71-s + 73-s − 77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.709867307\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.709867307\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
good | 2 | \( ( 1 - T )( 1 + T ) \) |
| 7 | \( 1 - T + T^{2} \) |
| 11 | \( 1 + T + T^{2} \) |
| 17 | \( 1 - T + T^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 + T )^{2} \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 + T )^{2} \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( 1 - T + T^{2} \) |
| 59 | \( 1 + T + T^{2} \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( 1 - T + T^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( 1 - T + T^{2} \) |
| 79 | \( 1 + T + T^{2} \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 - T )^{2} \) |
| 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
−9.806332239347547502979196540763, −8.541053271056987043184220697867, −7.81067699067325877581950450395, −7.23831680178860592325648898317, −6.19639916499991672544772905762, −5.46123467202962081157429579464, −4.87320675653006170690019699709, −3.34821931909363574218072871144, −2.26804105375163801907472717510, −1.70549595820603313776656871975,
1.70549595820603313776656871975, 2.26804105375163801907472717510, 3.34821931909363574218072871144, 4.87320675653006170690019699709, 5.46123467202962081157429579464, 6.19639916499991672544772905762, 7.23831680178860592325648898317, 7.81067699067325877581950450395, 8.541053271056987043184220697867, 9.806332239347547502979196540763