L(s) = 1 | + (0.707 − 0.707i)2-s + (0.707 − 0.707i)3-s − 1.00i·4-s − 1.00i·6-s + (−0.707 − 0.707i)8-s − 1.00i·9-s + (−0.707 − 0.707i)12-s − 1.00·16-s + 1.41i·17-s + (−0.707 − 0.707i)18-s + (1 + i)19-s − 1.41·23-s − 1.00·24-s + (−0.707 − 0.707i)27-s + (−0.707 + 0.707i)32-s + ⋯ |
L(s) = 1 | + (0.707 − 0.707i)2-s + (0.707 − 0.707i)3-s − 1.00i·4-s − 1.00i·6-s + (−0.707 − 0.707i)8-s − 1.00i·9-s + (−0.707 − 0.707i)12-s − 1.00·16-s + 1.41i·17-s + (−0.707 − 0.707i)18-s + (1 + i)19-s − 1.41·23-s − 1.00·24-s + (−0.707 − 0.707i)27-s + (−0.707 + 0.707i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.817400993\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.817400993\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 3 | \( 1 + (-0.707 + 0.707i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - T^{2} \) |
| 11 | \( 1 + iT^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 - 1.41iT - T^{2} \) |
| 19 | \( 1 + (-1 - i)T + iT^{2} \) |
| 23 | \( 1 + 1.41T + T^{2} \) |
| 29 | \( 1 - iT^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + 1.41iT - T^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 + iT^{2} \) |
| 61 | \( 1 + (-1 - i)T + iT^{2} \) |
| 67 | \( 1 + iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + 2T + T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−9.941308522609251798665432272118, −8.873904816791285466931594360543, −8.123330196420296618394460095524, −7.17941249348476041711943969770, −6.16401463795883761344139401552, −5.57226206177328113976871753949, −4.08158034082665512924573218032, −3.52839064144937048498977292805, −2.32311407062823274440728446077, −1.41653257128971909545445554546,
2.44930427984677945439635248465, 3.24688955565716854955685822371, 4.27611281541011731829627389402, 4.99501247829726113877168130486, 5.82370807001801801500211143230, 7.05560212437867572219302770829, 7.63820609103880437956876881908, 8.503495125939194254862027939810, 9.301993950216402422605559432622, 9.911921954395199987151131381259