| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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
−10.95789202929013331604096528050, −9.428551215501720449718838795900, −8.939320923146937509086992062761, −8.072850222505667551556927881472, −6.77545788870985340021630699596, −6.03555047376347676102488392350, −4.20618639180460077621153457505, −3.45746702875020591393751882693, −2.07758667987866208522623721440, 0,
2.07758667987866208522623721440, 3.45746702875020591393751882693, 4.20618639180460077621153457505, 6.03555047376347676102488392350, 6.77545788870985340021630699596, 8.072850222505667551556927881472, 8.939320923146937509086992062761, 9.428551215501720449718838795900, 10.95789202929013331604096528050
