L(s) = 1 | + (1.34 + 1.34i)2-s + (−0.965 + 0.399i)3-s + 2.61i·4-s + (−1.83 − 0.760i)6-s + (0.0600 − 0.144i)7-s + (−2.17 + 2.17i)8-s + (0.0650 − 0.0650i)9-s + (−1.04 − 2.52i)12-s + (0.275 − 0.114i)14-s − 3.23·16-s + (0.809 − 0.587i)17-s + 0.175·18-s + 0.163i·21-s + (1.23 − 2.97i)24-s + (−0.707 + 0.707i)25-s + ⋯ |
L(s) = 1 | + (1.34 + 1.34i)2-s + (−0.965 + 0.399i)3-s + 2.61i·4-s + (−1.83 − 0.760i)6-s + (0.0600 − 0.144i)7-s + (−2.17 + 2.17i)8-s + (0.0650 − 0.0650i)9-s + (−1.04 − 2.52i)12-s + (0.275 − 0.114i)14-s − 3.23·16-s + (0.809 − 0.587i)17-s + 0.175·18-s + 0.163i·21-s + (1.23 − 2.97i)24-s + (−0.707 + 0.707i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 - 0.202i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 - 0.202i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.341206277\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.341206277\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 + (-0.809 + 0.587i)T \) |
| 47 | \( 1 - iT \) |
good | 2 | \( 1 + (-1.34 - 1.34i)T + iT^{2} \) |
| 3 | \( 1 + (0.965 - 0.399i)T + (0.707 - 0.707i)T^{2} \) |
| 5 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 7 | \( 1 + (-0.0600 + 0.144i)T + (-0.707 - 0.707i)T^{2} \) |
| 11 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 19 | \( 1 - iT^{2} \) |
| 23 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 29 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 31 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 37 | \( 1 + (-1.57 + 0.652i)T + (0.707 - 0.707i)T^{2} \) |
| 41 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 53 | \( 1 + (-0.642 - 0.642i)T + iT^{2} \) |
| 59 | \( 1 + (-0.437 + 0.437i)T - iT^{2} \) |
| 61 | \( 1 + (-0.581 + 1.40i)T + (-0.707 - 0.707i)T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (1.79 - 0.744i)T + (0.707 - 0.707i)T^{2} \) |
| 73 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 79 | \( 1 + (1.79 + 0.744i)T + (0.707 + 0.707i)T^{2} \) |
| 83 | \( 1 + (1 + i)T + iT^{2} \) |
| 89 | \( 1 + 1.90iT - T^{2} \) |
| 97 | \( 1 + (-0.497 - 1.20i)T + (-0.707 + 0.707i)T^{2} \) |
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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
−11.32135055946026197356191293527, −10.09217958243188944582337734021, −8.941413934642124297844309709739, −7.80077280664872740052851061414, −7.28138900562112459681604797370, −6.08404415854813004472361907119, −5.69683521035988789844062148963, −4.81461616176181746872550031763, −4.07980234028456895739692246494, −2.88335228674321957833288608823,
1.12032558219798655196954206314, 2.43978365373120494013713083807, 3.61626603399814769838508462041, 4.56822610504917698250411205351, 5.65480632060527779831597570545, 5.95897638871442382917198102240, 7.04293820468140210496215796913, 8.559606026577655924193663101755, 9.859736775048510603699295800498, 10.35092997999824597973500051509