L(s) = 1 | + 5-s + 7-s − 2·11-s + 6·17-s + 4·19-s + 23-s + 25-s − 2·29-s − 10·31-s + 35-s − 8·37-s − 6·41-s + 12·43-s + 4·47-s + 49-s + 8·53-s − 2·55-s − 10·59-s + 6·61-s + 8·67-s + 4·71-s − 16·73-s − 2·77-s − 10·79-s − 12·83-s + 6·85-s + 6·89-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 0.377·7-s − 0.603·11-s + 1.45·17-s + 0.917·19-s + 0.208·23-s + 1/5·25-s − 0.371·29-s − 1.79·31-s + 0.169·35-s − 1.31·37-s − 0.937·41-s + 1.82·43-s + 0.583·47-s + 1/7·49-s + 1.09·53-s − 0.269·55-s − 1.30·59-s + 0.768·61-s + 0.977·67-s + 0.474·71-s − 1.87·73-s − 0.227·77-s − 1.12·79-s − 1.31·83-s + 0.650·85-s + 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−13.94756485885724, −13.32150716995425, −12.92589481099509, −12.31045714073213, −12.07058546645728, −11.33533647378241, −10.94529971564378, −10.38948989728338, −9.961302319611004, −9.544384484493789, −8.774605153854453, −8.647975993184194, −7.717655481464691, −7.357127195732895, −7.158600375453582, −6.152661810648403, −5.613234370466562, −5.372427529732366, −4.876141506285254, −3.990502464375432, −3.503559494232171, −2.918177474012922, −2.242637491879188, −1.552281449621212, −0.9967402506326371, 0,
0.9967402506326371, 1.552281449621212, 2.242637491879188, 2.918177474012922, 3.503559494232171, 3.990502464375432, 4.876141506285254, 5.372427529732366, 5.613234370466562, 6.152661810648403, 7.158600375453582, 7.357127195732895, 7.717655481464691, 8.647975993184194, 8.774605153854453, 9.544384484493789, 9.961302319611004, 10.38948989728338, 10.94529971564378, 11.33533647378241, 12.07058546645728, 12.31045714073213, 12.92589481099509, 13.32150716995425, 13.94756485885724