| L(s) = 1 | − i·2-s + 1.63i·3-s − 4-s + 1.63·6-s + i·8-s + 0.316·9-s − 1.31·11-s − 1.63i·12-s + 6.10i·13-s + 16-s − 2.60i·17-s − 0.316i·18-s + 3.05·19-s + 1.31i·22-s + 4.63i·23-s − 1.63·24-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.945i·3-s − 0.5·4-s + 0.668·6-s + 0.353i·8-s + 0.105·9-s − 0.396·11-s − 0.472i·12-s + 1.69i·13-s + 0.250·16-s − 0.631i·17-s − 0.0746i·18-s + 0.700·19-s + 0.280i·22-s + 0.966i·23-s − 0.334·24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.047001109\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.047001109\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 1.63iT - 3T^{2} \) |
| 11 | \( 1 + 1.31T + 11T^{2} \) |
| 13 | \( 1 - 6.10iT - 13T^{2} \) |
| 17 | \( 1 + 2.60iT - 17T^{2} \) |
| 19 | \( 1 - 3.05T + 19T^{2} \) |
| 23 | \( 1 - 4.63iT - 23T^{2} \) |
| 29 | \( 1 + 10.6T + 29T^{2} \) |
| 31 | \( 1 + 5.65T + 31T^{2} \) |
| 37 | \( 1 + 8.63iT - 37T^{2} \) |
| 41 | \( 1 - 7.29T + 41T^{2} \) |
| 43 | \( 1 + 6.63iT - 43T^{2} \) |
| 47 | \( 1 - 3.27iT - 47T^{2} \) |
| 53 | \( 1 - 8iT - 53T^{2} \) |
| 59 | \( 1 - 7.51T + 59T^{2} \) |
| 61 | \( 1 + 3.27T + 61T^{2} \) |
| 67 | \( 1 - 11.6iT - 67T^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 - 2.60iT - 73T^{2} \) |
| 79 | \( 1 + 14.6T + 79T^{2} \) |
| 83 | \( 1 - 12.9iT - 83T^{2} \) |
| 89 | \( 1 - 9.15T + 89T^{2} \) |
| 97 | \( 1 - 4.24iT - 97T^{2} \) |
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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
−9.213803430171740899793199810315, −9.047584219005684803907577660677, −7.52091904626849351869215141387, −7.16244365925782083486667234901, −5.69393053161109265349370719067, −5.15369654785364491040620121447, −4.03420391644567441427692538829, −3.82786484733543656576139081204, −2.51352939902224778961264966075, −1.49931512762928243414905154229,
0.35557509223734071110619820029, 1.58265214642154263315661051265, 2.84429898283260598880298826356, 3.85218885816045756267470548423, 5.01827227889543370133773250972, 5.72465237136486188958319192175, 6.40788793026687695417457979046, 7.31958638456405793681486969438, 7.80240938555716332347595037037, 8.303044641184410369425903801865
