L(s) = 1 | + (−0.707 + 0.707i)3-s + (1.41 + 1.41i)7-s − 1.00i·9-s − 2.00·21-s + (0.707 + 0.707i)27-s + (−1.41 + 1.41i)43-s + 3.00i·49-s + 2·61-s + (1.41 − 1.41i)63-s + (−1.41 − 1.41i)67-s − 1.00·81-s + (1.41 − 1.41i)103-s − 2i·109-s + ⋯ |
L(s) = 1 | + (−0.707 + 0.707i)3-s + (1.41 + 1.41i)7-s − 1.00i·9-s − 2.00·21-s + (0.707 + 0.707i)27-s + (−1.41 + 1.41i)43-s + 3.00i·49-s + 2·61-s + (1.41 − 1.41i)63-s + (−1.41 − 1.41i)67-s − 1.00·81-s + (1.41 − 1.41i)103-s − 2i·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9523646389\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9523646389\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.707 - 0.707i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-1.41 - 1.41i)T + iT^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 + iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + iT^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + (1.41 - 1.41i)T - iT^{2} \) |
| 47 | \( 1 - iT^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - 2T + T^{2} \) |
| 67 | \( 1 + (1.41 + 1.41i)T + iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 - iT^{2} \) |
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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
−10.14613023945727100520028525829, −9.288478805347731116282825412905, −8.596412715270400328906903894697, −7.85466594608480222611769401287, −6.57738372855317584120674119835, −5.69567846077450316822158419440, −5.07473701428459983573087400675, −4.35211579153683385784979435733, −2.98876520402881796355534311222, −1.67850746664169899658497711978,
1.03672170738375410860717161310, 2.07364776671274661445798946880, 3.81653063637242672878343513336, 4.76270548427746030080657825137, 5.43690937806494213244648617856, 6.65009205907508451557026934974, 7.29179208913815828792808368247, 7.943254919550881968610540621792, 8.690326971760936546789527163827, 10.21440078686898351557027584875