| L(s) = 1 | − 26·7-s − 45·11-s − 44·13-s + 117·17-s − 91·19-s − 18·23-s − 144·29-s + 26·31-s + 214·37-s + 459·41-s + 460·43-s − 468·47-s + 333·49-s + 558·53-s + 72·59-s − 118·61-s − 251·67-s − 108·71-s − 299·73-s + 1.17e3·77-s − 898·79-s + 927·83-s − 351·89-s + 1.14e3·91-s − 386·97-s + 954·101-s + 772·103-s + ⋯ |
| L(s) = 1 | − 1.40·7-s − 1.23·11-s − 0.938·13-s + 1.66·17-s − 1.09·19-s − 0.163·23-s − 0.922·29-s + 0.150·31-s + 0.950·37-s + 1.74·41-s + 1.63·43-s − 1.45·47-s + 0.970·49-s + 1.44·53-s + 0.158·59-s − 0.247·61-s − 0.457·67-s − 0.180·71-s − 0.479·73-s + 1.73·77-s − 1.27·79-s + 1.22·83-s − 0.418·89-s + 1.31·91-s − 0.404·97-s + 0.939·101-s + 0.738·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.037488634\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.037488634\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 45 T + p^{3} T^{2} \) |
| 13 | \( 1 + 44 T + p^{3} T^{2} \) |
| 17 | \( 1 - 117 T + p^{3} T^{2} \) |
| 19 | \( 1 + 91 T + p^{3} T^{2} \) |
| 23 | \( 1 + 18 T + p^{3} T^{2} \) |
| 29 | \( 1 + 144 T + p^{3} T^{2} \) |
| 31 | \( 1 - 26 T + p^{3} T^{2} \) |
| 37 | \( 1 - 214 T + p^{3} T^{2} \) |
| 41 | \( 1 - 459 T + p^{3} T^{2} \) |
| 43 | \( 1 - 460 T + p^{3} T^{2} \) |
| 47 | \( 1 + 468 T + p^{3} T^{2} \) |
| 53 | \( 1 - 558 T + p^{3} T^{2} \) |
| 59 | \( 1 - 72 T + p^{3} T^{2} \) |
| 61 | \( 1 + 118 T + p^{3} T^{2} \) |
| 67 | \( 1 + 251 T + p^{3} T^{2} \) |
| 71 | \( 1 + 108 T + p^{3} T^{2} \) |
| 73 | \( 1 + 299 T + p^{3} T^{2} \) |
| 79 | \( 1 + 898 T + p^{3} T^{2} \) |
| 83 | \( 1 - 927 T + p^{3} T^{2} \) |
| 89 | \( 1 + 351 T + p^{3} T^{2} \) |
| 97 | \( 1 + 386 T + p^{3} T^{2} \) |
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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
−9.893182287223761317928110142432, −9.042791416320959893474468464633, −7.83387189514703590099354327069, −7.33256492330682363581493375172, −6.13732291117610307097275765505, −5.52831872335891968864388407059, −4.30512446376186688418637641867, −3.16200603343208551791067142070, −2.37796005516058082757505168548, −0.52490410389325957252324575297,
0.52490410389325957252324575297, 2.37796005516058082757505168548, 3.16200603343208551791067142070, 4.30512446376186688418637641867, 5.52831872335891968864388407059, 6.13732291117610307097275765505, 7.33256492330682363581493375172, 7.83387189514703590099354327069, 9.042791416320959893474468464633, 9.893182287223761317928110142432
