| L(s) = 1 | − 3-s + 5-s + 9-s − 15-s − 17-s − 4·19-s + 25-s − 27-s + 10·29-s + 8·31-s + 6·37-s + 2·41-s − 8·43-s + 45-s − 7·49-s + 51-s + 2·53-s + 4·57-s + 12·59-s + 14·61-s + 8·67-s − 4·71-s − 6·73-s − 75-s − 16·79-s + 81-s − 4·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1/3·9-s − 0.258·15-s − 0.242·17-s − 0.917·19-s + 1/5·25-s − 0.192·27-s + 1.85·29-s + 1.43·31-s + 0.986·37-s + 0.312·41-s − 1.21·43-s + 0.149·45-s − 49-s + 0.140·51-s + 0.274·53-s + 0.529·57-s + 1.56·59-s + 1.79·61-s + 0.977·67-s − 0.474·71-s − 0.702·73-s − 0.115·75-s − 1.80·79-s + 1/9·81-s − 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 344760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 344760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 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
−12.86954194409825, −12.30570592510720, −11.74120152467887, −11.56205461858021, −11.01616192133241, −10.36124913071808, −10.15134299927148, −9.813450662155212, −9.177656897984423, −8.575137019653510, −8.243953792254405, −7.884571155411251, −6.950346104832589, −6.654608733860499, −6.453785643960858, −5.742347554346865, −5.356517126327293, −4.725107646047041, −4.375200827209515, −3.893911301487894, −3.030546677024564, −2.601276660706163, −2.075052187611397, −1.270769282936432, −0.8281720528708248, 0,
0.8281720528708248, 1.270769282936432, 2.075052187611397, 2.601276660706163, 3.030546677024564, 3.893911301487894, 4.375200827209515, 4.725107646047041, 5.356517126327293, 5.742347554346865, 6.453785643960858, 6.654608733860499, 6.950346104832589, 7.884571155411251, 8.243953792254405, 8.575137019653510, 9.177656897984423, 9.813450662155212, 10.15134299927148, 10.36124913071808, 11.01616192133241, 11.56205461858021, 11.74120152467887, 12.30570592510720, 12.86954194409825
