| L(s) = 1 | − 3-s + 4·7-s + 9-s − 2·11-s − 13-s + 2·19-s − 4·21-s − 2·23-s − 27-s + 10·29-s + 4·31-s + 2·33-s + 6·37-s + 39-s − 6·41-s + 8·43-s + 12·47-s + 9·49-s + 14·53-s − 2·57-s + 6·59-s + 2·61-s + 4·63-s − 4·67-s + 2·69-s − 14·73-s − 8·77-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.51·7-s + 1/3·9-s − 0.603·11-s − 0.277·13-s + 0.458·19-s − 0.872·21-s − 0.417·23-s − 0.192·27-s + 1.85·29-s + 0.718·31-s + 0.348·33-s + 0.986·37-s + 0.160·39-s − 0.937·41-s + 1.21·43-s + 1.75·47-s + 9/7·49-s + 1.92·53-s − 0.264·57-s + 0.781·59-s + 0.256·61-s + 0.503·63-s − 0.488·67-s + 0.240·69-s − 1.63·73-s − 0.911·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.544537820\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.544537820\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−15.17921961560862, −14.45473761669562, −14.13131165163724, −13.53498410006579, −13.01480952635478, −12.12591600848690, −11.94364803754922, −11.47554694947130, −10.76405367410276, −10.36348738413076, −9.947246756273973, −9.065725445912415, −8.390307476236617, −8.106476379345812, −7.296795658763661, −7.044060848327740, −5.939763394194235, −5.689826436355988, −4.855543454062475, −4.565589591644412, −3.908384793248021, −2.742579096913880, −2.303196884584353, −1.284410597883130, −0.7037483478066092,
0.7037483478066092, 1.284410597883130, 2.303196884584353, 2.742579096913880, 3.908384793248021, 4.565589591644412, 4.855543454062475, 5.689826436355988, 5.939763394194235, 7.044060848327740, 7.296795658763661, 8.106476379345812, 8.390307476236617, 9.065725445912415, 9.947246756273973, 10.36348738413076, 10.76405367410276, 11.47554694947130, 11.94364803754922, 12.12591600848690, 13.01480952635478, 13.53498410006579, 14.13131165163724, 14.45473761669562, 15.17921961560862
