| L(s) = 1 | − 4.42·5-s − 3·9-s + 3.52i·11-s + 5.32i·19-s − 4.79i·23-s + 14.5·25-s − 9.59i·31-s − 2.61·37-s + 9.59·41-s − 12.3i·43-s + 13.2·45-s + 2i·47-s − 7·49-s − 6.23·53-s − 15.5i·55-s + ⋯ |
| L(s) = 1 | − 1.97·5-s − 9-s + 1.06i·11-s + 1.22i·19-s − 0.999i·23-s + 2.91·25-s − 1.72i·31-s − 0.430·37-s + 1.49·41-s − 1.88i·43-s + 1.97·45-s + 0.291i·47-s − 49-s − 0.856·53-s − 2.10i·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6800989542\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6800989542\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 + 4.79iT \) |
| good | 3 | \( 1 + 3T^{2} \) |
| 5 | \( 1 + 4.42T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 - 3.52iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 5.32iT - 19T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 9.59iT - 31T^{2} \) |
| 37 | \( 1 + 2.61T + 37T^{2} \) |
| 41 | \( 1 - 9.59T + 41T^{2} \) |
| 43 | \( 1 + 12.3iT - 43T^{2} \) |
| 47 | \( 1 - 2iT - 47T^{2} \) |
| 53 | \( 1 + 6.23T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 11.4T + 61T^{2} \) |
| 67 | \( 1 + 14.1iT - 67T^{2} \) |
| 71 | \( 1 - 10iT - 71T^{2} \) |
| 73 | \( 1 - 9.59T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 1.71iT - 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−9.271851530869613368753884931048, −8.346034933422586139459311271773, −7.900042632925607526062213895625, −7.19629716236669054597983393412, −6.22543010934544776541259312737, −5.05758675158188326331067041340, −4.17441747258869258960202483439, −3.54278878327944923712326826428, −2.34683113271580161142778332825, −0.41440683688438305405515625671,
0.78545072692815569191357294212, 2.98040560144960501088618373655, 3.39727092282995342581525988359, 4.50385056698379284499245821831, 5.33341417165652305385673641027, 6.48622701954488997192543931769, 7.31395666309106153929649257445, 8.133415410740165666463873622797, 8.573755339857942359123968032375, 9.346404945013929528508741950504
