L(s) = 1 | + (0.707 + 0.707i)2-s + 1.00i·4-s + (1.70 − 0.707i)7-s + (−0.707 + 0.707i)8-s + (1.70 + 0.707i)14-s − 1.00·16-s − i·17-s + (−0.707 + 0.292i)23-s + (−0.707 − 0.707i)25-s + (0.707 + 1.70i)28-s + (−0.707 + 1.70i)31-s + (−0.707 − 0.707i)32-s + (0.707 − 0.707i)34-s + (0.707 + 1.70i)41-s + (−0.707 − 0.292i)46-s − 1.41·47-s + ⋯ |
L(s) = 1 | + (0.707 + 0.707i)2-s + 1.00i·4-s + (1.70 − 0.707i)7-s + (−0.707 + 0.707i)8-s + (1.70 + 0.707i)14-s − 1.00·16-s − i·17-s + (−0.707 + 0.292i)23-s + (−0.707 − 0.707i)25-s + (0.707 + 1.70i)28-s + (−0.707 + 1.70i)31-s + (−0.707 − 0.707i)32-s + (0.707 − 0.707i)34-s + (0.707 + 1.70i)41-s + (−0.707 − 0.292i)46-s − 1.41·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.513 - 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.513 - 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.702483346\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.702483346\) |
\(L(1)\) |
|
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 | \( 1 \) |
| 17 | \( 1 + iT \) |
good | 5 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 7 | \( 1 + (-1.70 + 0.707i)T + (0.707 - 0.707i)T^{2} \) |
| 11 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 19 | \( 1 + iT^{2} \) |
| 23 | \( 1 + (0.707 - 0.292i)T + (0.707 - 0.707i)T^{2} \) |
| 29 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 31 | \( 1 + (0.707 - 1.70i)T + (-0.707 - 0.707i)T^{2} \) |
| 37 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 41 | \( 1 + (-0.707 - 1.70i)T + (-0.707 + 0.707i)T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + 1.41T + T^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 + iT^{2} \) |
| 61 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (1.70 + 0.707i)T + (0.707 + 0.707i)T^{2} \) |
| 73 | \( 1 + (0.707 + 0.292i)T + (0.707 + 0.707i)T^{2} \) |
| 79 | \( 1 + (0.292 + 0.707i)T + (-0.707 + 0.707i)T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 - 1.41T + T^{2} \) |
| 97 | \( 1 + (0.707 + 0.292i)T + (0.707 + 0.707i)T^{2} \) |
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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
−10.09853589921631704620052454385, −8.955779551721176289336551690066, −8.094289855583212984763297607195, −7.62744963743802010497337797874, −6.82370450875603030219720714401, −5.74520584777506317499909352236, −4.82842798297936464632734958550, −4.37155640023847310752728970797, −3.16315848821509543028962822456, −1.74957326672877582174375742629,
1.64611876334853227501194092828, 2.31128585095778361805872346941, 3.79899646007313842851145514025, 4.52707859472229565265934182160, 5.53690399569354847245813687953, 5.97328851808963934472003935638, 7.39769421723455263637070589524, 8.249889994833035624905979481068, 9.032273034779138521086367405805, 9.963759920739648523989276983339