L(s) = 1 | + (−1 − i)2-s + i·4-s + (−0.707 + 0.707i)7-s − 1.41i·11-s + 1.41·14-s + 16-s + (−1.41 + 1.41i)22-s + (1 − i)23-s + (−0.707 − 0.707i)28-s − 1.41·29-s + (−1 − i)32-s + (1.41 − 1.41i)37-s + (−1.41 − 1.41i)43-s + 1.41·44-s − 2·46-s + ⋯ |
L(s) = 1 | + (−1 − i)2-s + i·4-s + (−0.707 + 0.707i)7-s − 1.41i·11-s + 1.41·14-s + 16-s + (−1.41 + 1.41i)22-s + (1 − i)23-s + (−0.707 − 0.707i)28-s − 1.41·29-s + (−1 − i)32-s + (1.41 − 1.41i)37-s + (−1.41 − 1.41i)43-s + 1.41·44-s − 2·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 + 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 + 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4799929473\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4799929473\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (0.707 - 0.707i)T \) |
good | 2 | \( 1 + (1 + i)T + iT^{2} \) |
| 11 | \( 1 + 1.41iT - T^{2} \) |
| 13 | \( 1 - iT^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + (-1 + i)T - iT^{2} \) |
| 29 | \( 1 + 1.41T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (-1.41 + 1.41i)T - iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + (1.41 + 1.41i)T + iT^{2} \) |
| 47 | \( 1 - iT^{2} \) |
| 53 | \( 1 + (-1 + i)T - iT^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 - 1.41iT - T^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 + 2iT - T^{2} \) |
| 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
−9.242454152782392599432558068946, −8.809406131187633599072289804249, −8.174510003257792654563353136146, −7.03389790411142241419526366732, −5.98411915569158143543096113180, −5.38058519732826601623193750417, −3.75531217525883219134556207398, −2.98147865358503158802669113073, −2.11365510750594449218689920851, −0.56453713107814773700230536141,
1.35851116374147052679949178726, 3.03114450872281549054864803105, 4.12497499643371629024693422102, 5.19765565831339564062725644564, 6.26832763004306172893859986984, 6.94340333399094639314802054835, 7.48490087090307956312785115857, 8.134487668699395820375558157312, 9.382041697360766707062629613688, 9.527287496820352143026150131037