L(s) = 1 | + 2·7-s − 2·11-s − 2·13-s + 6·17-s + 8·19-s − 4·23-s − 8·29-s + 10·37-s − 2·41-s + 12·43-s − 3·49-s + 10·53-s + 6·59-s + 2·61-s + 8·67-s + 4·71-s − 4·73-s − 4·77-s − 8·79-s + 4·83-s − 6·89-s − 4·91-s − 8·97-s + 2·103-s + 4·107-s − 6·109-s + 2·113-s + ⋯ |
L(s) = 1 | + 0.755·7-s − 0.603·11-s − 0.554·13-s + 1.45·17-s + 1.83·19-s − 0.834·23-s − 1.48·29-s + 1.64·37-s − 0.312·41-s + 1.82·43-s − 3/7·49-s + 1.37·53-s + 0.781·59-s + 0.256·61-s + 0.977·67-s + 0.474·71-s − 0.468·73-s − 0.455·77-s − 0.900·79-s + 0.439·83-s − 0.635·89-s − 0.419·91-s − 0.812·97-s + 0.197·103-s + 0.386·107-s − 0.574·109-s + 0.188·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.902627256\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.902627256\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−9.552205182610048147468552978643, −8.276305178538120606802319498114, −7.66954264780705476008575525598, −7.22831951971231942039552415252, −5.66821441984527028491370693629, −5.46409672499685018038392724670, −4.34191555410904899299129102689, −3.31946416110453543237655105934, −2.26633293984614676346934739749, −0.984372882186715294156472332258,
0.984372882186715294156472332258, 2.26633293984614676346934739749, 3.31946416110453543237655105934, 4.34191555410904899299129102689, 5.46409672499685018038392724670, 5.66821441984527028491370693629, 7.22831951971231942039552415252, 7.66954264780705476008575525598, 8.276305178538120606802319498114, 9.552205182610048147468552978643