L(s) = 1 | + 3-s + (0.707 + 0.707i)5-s + (−0.707 − 0.707i)7-s + 9-s + (1 + i)13-s + (0.707 + 0.707i)15-s + (0.707 − 0.707i)17-s + (−0.707 − 0.707i)21-s + (−0.707 − 0.707i)23-s + 1.00i·25-s + 27-s − 29-s − 31-s − 1.00i·35-s + (0.707 + 0.707i)37-s + ⋯ |
L(s) = 1 | + 3-s + (0.707 + 0.707i)5-s + (−0.707 − 0.707i)7-s + 9-s + (1 + i)13-s + (0.707 + 0.707i)15-s + (0.707 − 0.707i)17-s + (−0.707 − 0.707i)21-s + (−0.707 − 0.707i)23-s + 1.00i·25-s + 27-s − 29-s − 31-s − 1.00i·35-s + (0.707 + 0.707i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.937396787\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.937396787\) |
\(L(1)\) |
|
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 | \( 1 + (-0.707 - 0.707i)T \) |
| 23 | \( 1 + (0.707 + 0.707i)T \) |
good | 7 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + (-1 - i)T + iT^{2} \) |
| 17 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 29 | \( 1 + T + T^{2} \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( 1 + (-0.707 - 0.707i)T + iT^{2} \) |
| 41 | \( 1 + iT - T^{2} \) |
| 43 | \( 1 + (1.41 - 1.41i)T - iT^{2} \) |
| 47 | \( 1 + iT^{2} \) |
| 53 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
| 59 | \( 1 - T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + (-0.707 - 0.707i)T + iT^{2} \) |
| 71 | \( 1 + iT - T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 - 1.41T + T^{2} \) |
| 83 | \( 1 + (-0.707 - 0.707i)T + iT^{2} \) |
| 89 | \( 1 + 1.41iT - T^{2} \) |
| 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
−9.245293332263819504950130078317, −8.267906454512528224024052207678, −7.47873051358083040087860395888, −6.72570425030256007326997903686, −6.30800836710070171479535163755, −5.11042990642669460957400026336, −3.87218810985950202495635614610, −3.47713980513251299279468740598, −2.44095352682031156556609700009, −1.49355093912207211627991245028,
1.38055526986669349863331196469, 2.28687836999591453129113285420, 3.36296453139451773464353957715, 3.91837664815999178445824907671, 5.31215820555109293747284713240, 5.79279385219315183569770941731, 6.60418449535719205467423978239, 7.81294193654265839512179551305, 8.231614246038479565123458309665, 9.065389966801100853673169932786