| L(s) = 1 | + 4·3-s + 5·5-s + 16·7-s − 11·9-s + 36·11-s − 42·13-s + 20·15-s − 110·17-s − 116·19-s + 64·21-s + 16·23-s + 25·25-s − 152·27-s + 198·29-s + 240·31-s + 144·33-s + 80·35-s − 258·37-s − 168·39-s + 442·41-s − 292·43-s − 55·45-s + 392·47-s − 87·49-s − 440·51-s + 142·53-s + 180·55-s + ⋯ |
| L(s) = 1 | + 0.769·3-s + 0.447·5-s + 0.863·7-s − 0.407·9-s + 0.986·11-s − 0.896·13-s + 0.344·15-s − 1.56·17-s − 1.40·19-s + 0.665·21-s + 0.145·23-s + 1/5·25-s − 1.08·27-s + 1.26·29-s + 1.39·31-s + 0.759·33-s + 0.386·35-s − 1.14·37-s − 0.689·39-s + 1.68·41-s − 1.03·43-s − 0.182·45-s + 1.21·47-s − 0.253·49-s − 1.20·51-s + 0.368·53-s + 0.441·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.677307176\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.677307176\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 4 T + p^{3} T^{2} \) |
| 7 | \( 1 - 16 T + p^{3} T^{2} \) |
| 11 | \( 1 - 36 T + p^{3} T^{2} \) |
| 13 | \( 1 + 42 T + p^{3} T^{2} \) |
| 17 | \( 1 + 110 T + p^{3} T^{2} \) |
| 19 | \( 1 + 116 T + p^{3} T^{2} \) |
| 23 | \( 1 - 16 T + p^{3} T^{2} \) |
| 29 | \( 1 - 198 T + p^{3} T^{2} \) |
| 31 | \( 1 - 240 T + p^{3} T^{2} \) |
| 37 | \( 1 + 258 T + p^{3} T^{2} \) |
| 41 | \( 1 - 442 T + p^{3} T^{2} \) |
| 43 | \( 1 + 292 T + p^{3} T^{2} \) |
| 47 | \( 1 - 392 T + p^{3} T^{2} \) |
| 53 | \( 1 - 142 T + p^{3} T^{2} \) |
| 59 | \( 1 + 348 T + p^{3} T^{2} \) |
| 61 | \( 1 + 570 T + p^{3} T^{2} \) |
| 67 | \( 1 - 692 T + p^{3} T^{2} \) |
| 71 | \( 1 - 168 T + p^{3} T^{2} \) |
| 73 | \( 1 + 134 T + p^{3} T^{2} \) |
| 79 | \( 1 - 784 T + p^{3} T^{2} \) |
| 83 | \( 1 - 564 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1034 T + p^{3} T^{2} \) |
| 97 | \( 1 + 382 T + p^{3} T^{2} \) |
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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
−15.31208157689612972873729431538, −14.44930236357312692398111275609, −13.64603356891853479737645757108, −12.10068595009826496479674508400, −10.79072822068917129893808373089, −9.193496150539753633078912038086, −8.293186819103647794666715126034, −6.54660529947676692424737057589, −4.54075474162946091907136991397, −2.29672964307359539613824771035,
2.29672964307359539613824771035, 4.54075474162946091907136991397, 6.54660529947676692424737057589, 8.293186819103647794666715126034, 9.193496150539753633078912038086, 10.79072822068917129893808373089, 12.10068595009826496479674508400, 13.64603356891853479737645757108, 14.44930236357312692398111275609, 15.31208157689612972873729431538
