| L(s) = 1 | + 3.25·3-s − 0.0778·7-s + 7.58·9-s + 4.50·11-s + 5.33·13-s − 7.33·17-s − 19-s − 0.253·21-s + 3.40·23-s + 14.9·27-s − 1.33·29-s + 2.50·31-s + 14.6·33-s + 5.50·37-s + 17.3·39-s + 0.506·43-s − 5.66·47-s − 6.99·49-s − 23.8·51-s + 12.9·53-s − 3.25·57-s − 7.56·59-s − 2.15·61-s − 0.590·63-s − 4.58·67-s + 11.0·69-s + 10.8·71-s + ⋯ |
| L(s) = 1 | + 1.87·3-s − 0.0294·7-s + 2.52·9-s + 1.35·11-s + 1.47·13-s − 1.77·17-s − 0.229·19-s − 0.0553·21-s + 0.710·23-s + 2.87·27-s − 0.247·29-s + 0.450·31-s + 2.55·33-s + 0.905·37-s + 2.77·39-s + 0.0772·43-s − 0.825·47-s − 0.999·49-s − 3.33·51-s + 1.77·53-s − 0.430·57-s − 0.984·59-s − 0.276·61-s − 0.0744·63-s − 0.560·67-s + 1.33·69-s + 1.28·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.228882922\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.228882922\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 3.25T + 3T^{2} \) |
| 7 | \( 1 + 0.0778T + 7T^{2} \) |
| 11 | \( 1 - 4.50T + 11T^{2} \) |
| 13 | \( 1 - 5.33T + 13T^{2} \) |
| 17 | \( 1 + 7.33T + 17T^{2} \) |
| 23 | \( 1 - 3.40T + 23T^{2} \) |
| 29 | \( 1 + 1.33T + 29T^{2} \) |
| 31 | \( 1 - 2.50T + 31T^{2} \) |
| 37 | \( 1 - 5.50T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 0.506T + 43T^{2} \) |
| 47 | \( 1 + 5.66T + 47T^{2} \) |
| 53 | \( 1 - 12.9T + 53T^{2} \) |
| 59 | \( 1 + 7.56T + 59T^{2} \) |
| 61 | \( 1 + 2.15T + 61T^{2} \) |
| 67 | \( 1 + 4.58T + 67T^{2} \) |
| 71 | \( 1 - 10.8T + 71T^{2} \) |
| 73 | \( 1 - 5.09T + 73T^{2} \) |
| 79 | \( 1 + 17.0T + 79T^{2} \) |
| 83 | \( 1 + 13.1T + 83T^{2} \) |
| 89 | \( 1 - 15.0T + 89T^{2} \) |
| 97 | \( 1 + 7.67T + 97T^{2} \) |
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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
−8.254168008548413599732330340934, −7.16674211789930999513021098393, −6.70222589258510888026074837672, −6.07061274331131884090174885501, −4.61408911472296840297958317629, −4.12134459134540070830352789327, −3.52511469717857374162895772033, −2.75071729846993269224692758719, −1.87664807835665591449194449477, −1.16386140225448027144453339666,
1.16386140225448027144453339666, 1.87664807835665591449194449477, 2.75071729846993269224692758719, 3.52511469717857374162895772033, 4.12134459134540070830352789327, 4.61408911472296840297958317629, 6.07061274331131884090174885501, 6.70222589258510888026074837672, 7.16674211789930999513021098393, 8.254168008548413599732330340934
