| L(s) = 1 | + 3-s + 0.0132·7-s + 9-s − 1.16·11-s + 0.593·13-s − 2.49·17-s − 3.00·19-s + 0.0132·21-s + 6.70·23-s + 27-s − 9.99·29-s + 31-s − 1.16·33-s + 7.22·37-s + 0.593·39-s + 8.37·41-s − 10.7·43-s − 4.62·47-s − 6.99·49-s − 2.49·51-s − 11.3·53-s − 3.00·57-s − 9.36·59-s − 1.10·61-s + 0.0132·63-s − 2.28·67-s + 6.70·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.00501·7-s + 0.333·9-s − 0.351·11-s + 0.164·13-s − 0.604·17-s − 0.690·19-s + 0.00289·21-s + 1.39·23-s + 0.192·27-s − 1.85·29-s + 0.179·31-s − 0.203·33-s + 1.18·37-s + 0.0950·39-s + 1.30·41-s − 1.63·43-s − 0.674·47-s − 0.999·49-s − 0.348·51-s − 1.55·53-s − 0.398·57-s − 1.21·59-s − 0.141·61-s + 0.00167·63-s − 0.279·67-s + 0.807·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 7 | \( 1 - 0.0132T + 7T^{2} \) |
| 11 | \( 1 + 1.16T + 11T^{2} \) |
| 13 | \( 1 - 0.593T + 13T^{2} \) |
| 17 | \( 1 + 2.49T + 17T^{2} \) |
| 19 | \( 1 + 3.00T + 19T^{2} \) |
| 23 | \( 1 - 6.70T + 23T^{2} \) |
| 29 | \( 1 + 9.99T + 29T^{2} \) |
| 37 | \( 1 - 7.22T + 37T^{2} \) |
| 41 | \( 1 - 8.37T + 41T^{2} \) |
| 43 | \( 1 + 10.7T + 43T^{2} \) |
| 47 | \( 1 + 4.62T + 47T^{2} \) |
| 53 | \( 1 + 11.3T + 53T^{2} \) |
| 59 | \( 1 + 9.36T + 59T^{2} \) |
| 61 | \( 1 + 1.10T + 61T^{2} \) |
| 67 | \( 1 + 2.28T + 67T^{2} \) |
| 71 | \( 1 - 14.4T + 71T^{2} \) |
| 73 | \( 1 + 4.16T + 73T^{2} \) |
| 79 | \( 1 + 11.4T + 79T^{2} \) |
| 83 | \( 1 - 12.9T + 83T^{2} \) |
| 89 | \( 1 - 13.4T + 89T^{2} \) |
| 97 | \( 1 - 3.83T + 97T^{2} \) |
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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
−7.59466447305858683263680611045, −6.62991535970381810823374726607, −6.19445889106187758078702083351, −5.14043149016458175708656213064, −4.61662067297366259863635739990, −3.73871481760874365763048015630, −3.03754997312307758896916804874, −2.23595870737612502894875846503, −1.38250896992520753672072840288, 0,
1.38250896992520753672072840288, 2.23595870737612502894875846503, 3.03754997312307758896916804874, 3.73871481760874365763048015630, 4.61662067297366259863635739990, 5.14043149016458175708656213064, 6.19445889106187758078702083351, 6.62991535970381810823374726607, 7.59466447305858683263680611045
