| L(s) = 1 | − 26·7-s + 59·11-s − 28·13-s + 5·17-s + 109·19-s − 194·23-s + 32·29-s + 10·31-s + 198·37-s − 117·41-s − 388·43-s − 68·47-s + 333·49-s − 18·53-s − 392·59-s − 710·61-s + 253·67-s + 612·71-s + 549·73-s − 1.53e3·77-s + 414·79-s − 121·83-s + 81·89-s + 728·91-s + 1.50e3·97-s + 234·101-s + 1.17e3·103-s + ⋯ |
| L(s) = 1 | − 1.40·7-s + 1.61·11-s − 0.597·13-s + 0.0713·17-s + 1.31·19-s − 1.75·23-s + 0.204·29-s + 0.0579·31-s + 0.879·37-s − 0.445·41-s − 1.37·43-s − 0.211·47-s + 0.970·49-s − 0.0466·53-s − 0.864·59-s − 1.49·61-s + 0.461·67-s + 1.02·71-s + 0.880·73-s − 2.27·77-s + 0.589·79-s − 0.160·83-s + 0.0964·89-s + 0.838·91-s + 1.57·97-s + 0.230·101-s + 1.12·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.623182114\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.623182114\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 26 T + p^{3} T^{2} \) |
| 11 | \( 1 - 59 T + p^{3} T^{2} \) |
| 13 | \( 1 + 28 T + p^{3} T^{2} \) |
| 17 | \( 1 - 5 T + p^{3} T^{2} \) |
| 19 | \( 1 - 109 T + p^{3} T^{2} \) |
| 23 | \( 1 + 194 T + p^{3} T^{2} \) |
| 29 | \( 1 - 32 T + p^{3} T^{2} \) |
| 31 | \( 1 - 10 T + p^{3} T^{2} \) |
| 37 | \( 1 - 198 T + p^{3} T^{2} \) |
| 41 | \( 1 + 117 T + p^{3} T^{2} \) |
| 43 | \( 1 + 388 T + p^{3} T^{2} \) |
| 47 | \( 1 + 68 T + p^{3} T^{2} \) |
| 53 | \( 1 + 18 T + p^{3} T^{2} \) |
| 59 | \( 1 + 392 T + p^{3} T^{2} \) |
| 61 | \( 1 + 710 T + p^{3} T^{2} \) |
| 67 | \( 1 - 253 T + p^{3} T^{2} \) |
| 71 | \( 1 - 612 T + p^{3} T^{2} \) |
| 73 | \( 1 - 549 T + p^{3} T^{2} \) |
| 79 | \( 1 - 414 T + p^{3} T^{2} \) |
| 83 | \( 1 + 121 T + p^{3} T^{2} \) |
| 89 | \( 1 - 81 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1502 T + p^{3} T^{2} \) |
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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
−9.170474303194529578552590647186, −8.111302615699579225715583640660, −7.21921641606883845754540527517, −6.44418372106866791886892183739, −5.96641922761082019807857810914, −4.75251869095824657444713976859, −3.73323162285852791753673394707, −3.14786917416481041115075918618, −1.84442289015236427275908077816, −0.59519584277076253382780006874,
0.59519584277076253382780006874, 1.84442289015236427275908077816, 3.14786917416481041115075918618, 3.73323162285852791753673394707, 4.75251869095824657444713976859, 5.96641922761082019807857810914, 6.44418372106866791886892183739, 7.21921641606883845754540527517, 8.111302615699579225715583640660, 9.170474303194529578552590647186
