| L(s) = 1 | − 4·3-s + 5·5-s − 4·7-s + 7·9-s − 20·15-s + 16·21-s + 44·23-s + 25·25-s + 8·27-s + 22·29-s − 20·35-s + 62·41-s + 76·43-s + 35·45-s − 4·47-s − 33·49-s + 58·61-s − 28·63-s − 116·67-s − 176·69-s − 100·75-s − 95·81-s + 76·83-s − 88·87-s − 142·89-s − 122·101-s + 44·103-s + ⋯ |
| L(s) = 1 | − 4/3·3-s + 5-s − 4/7·7-s + 7/9·9-s − 4/3·15-s + 0.761·21-s + 1.91·23-s + 25-s + 8/27·27-s + 0.758·29-s − 4/7·35-s + 1.51·41-s + 1.76·43-s + 7/9·45-s − 0.0851·47-s − 0.673·49-s + 0.950·61-s − 4/9·63-s − 1.73·67-s − 2.55·69-s − 4/3·75-s − 1.17·81-s + 0.915·83-s − 1.01·87-s − 1.59·89-s − 1.20·101-s + 0.427·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.148506109\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.148506109\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - p T \) |
| good | 3 | \( 1 + 4 T + p^{2} T^{2} \) |
| 7 | \( 1 + 4 T + p^{2} T^{2} \) |
| 11 | \( ( 1 - p T )( 1 + p T ) \) |
| 13 | \( ( 1 - p T )( 1 + p T ) \) |
| 17 | \( ( 1 - p T )( 1 + p T ) \) |
| 19 | \( ( 1 - p T )( 1 + p T ) \) |
| 23 | \( 1 - 44 T + p^{2} T^{2} \) |
| 29 | \( 1 - 22 T + p^{2} T^{2} \) |
| 31 | \( ( 1 - p T )( 1 + p T ) \) |
| 37 | \( ( 1 - p T )( 1 + p T ) \) |
| 41 | \( 1 - 62 T + p^{2} T^{2} \) |
| 43 | \( 1 - 76 T + p^{2} T^{2} \) |
| 47 | \( 1 + 4 T + p^{2} T^{2} \) |
| 53 | \( ( 1 - p T )( 1 + p T ) \) |
| 59 | \( ( 1 - p T )( 1 + p T ) \) |
| 61 | \( 1 - 58 T + p^{2} T^{2} \) |
| 67 | \( 1 + 116 T + p^{2} T^{2} \) |
| 71 | \( ( 1 - p T )( 1 + p T ) \) |
| 73 | \( ( 1 - p T )( 1 + p T ) \) |
| 79 | \( ( 1 - p T )( 1 + p T ) \) |
| 83 | \( 1 - 76 T + p^{2} T^{2} \) |
| 89 | \( 1 + 142 T + p^{2} T^{2} \) |
| 97 | \( ( 1 - p T )( 1 + p T ) \) |
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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
−11.22740481788185262962926929259, −10.65194249813392486381546037289, −9.705331721091465996602595372962, −8.851543172372084760114678283652, −7.18348310758832146320401611479, −6.32929217586198263583330827206, −5.60942082451760237622222310996, −4.64997652390376790683674803158, −2.81288652388204018644061864416, −0.948009041906528905655733605405,
0.948009041906528905655733605405, 2.81288652388204018644061864416, 4.64997652390376790683674803158, 5.60942082451760237622222310996, 6.32929217586198263583330827206, 7.18348310758832146320401611479, 8.851543172372084760114678283652, 9.705331721091465996602595372962, 10.65194249813392486381546037289, 11.22740481788185262962926929259
