| L(s) = 1 | − 2-s + 3·3-s + 4-s − 3·6-s + 5·7-s − 8-s + 6·9-s − 4·11-s + 3·12-s + 13-s − 5·14-s + 16-s + 3·17-s − 6·18-s + 19-s + 15·21-s + 4·22-s − 7·23-s − 3·24-s − 26-s + 9·27-s + 5·28-s − 3·29-s − 2·31-s − 32-s − 12·33-s − 3·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.73·3-s + 1/2·4-s − 1.22·6-s + 1.88·7-s − 0.353·8-s + 2·9-s − 1.20·11-s + 0.866·12-s + 0.277·13-s − 1.33·14-s + 1/4·16-s + 0.727·17-s − 1.41·18-s + 0.229·19-s + 3.27·21-s + 0.852·22-s − 1.45·23-s − 0.612·24-s − 0.196·26-s + 1.73·27-s + 0.944·28-s − 0.557·29-s − 0.359·31-s − 0.176·32-s − 2.08·33-s − 0.514·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.439650405\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.439650405\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 7 | \( 1 - 5 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 13 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 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
−9.976858889227114824891335123219, −8.920485942669728815604036529685, −8.301058617244315182063927960266, −7.80203381755481970407982696068, −7.37288434104770513673343989750, −5.66402186780550341225871252054, −4.58979652776477679003498012022, −3.44171251094378542463371483956, −2.28599476561718030727276453764, −1.57691270891887807421558967069,
1.57691270891887807421558967069, 2.28599476561718030727276453764, 3.44171251094378542463371483956, 4.58979652776477679003498012022, 5.66402186780550341225871252054, 7.37288434104770513673343989750, 7.80203381755481970407982696068, 8.301058617244315182063927960266, 8.920485942669728815604036529685, 9.976858889227114824891335123219
