| L(s) = 1 | − 1.90i·2-s − 1.62·4-s − 4.42i·7-s − 0.719i·8-s − 11-s + 0.622i·13-s − 8.42·14-s − 4.61·16-s + 5.18i·17-s − 7.05·19-s + 1.90i·22-s + 8.85i·23-s + 1.18·26-s + 7.18i·28-s − 7.80·29-s + ⋯ |
| L(s) = 1 | − 1.34i·2-s − 0.811·4-s − 1.67i·7-s − 0.254i·8-s − 0.301·11-s + 0.172i·13-s − 2.25·14-s − 1.15·16-s + 1.25i·17-s − 1.61·19-s + 0.405i·22-s + 1.84i·23-s + 0.232·26-s + 1.35i·28-s − 1.44·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2478272938\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2478272938\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + 1.90iT - 2T^{2} \) |
| 7 | \( 1 + 4.42iT - 7T^{2} \) |
| 13 | \( 1 - 0.622iT - 13T^{2} \) |
| 17 | \( 1 - 5.18iT - 17T^{2} \) |
| 19 | \( 1 + 7.05T + 19T^{2} \) |
| 23 | \( 1 - 8.85iT - 23T^{2} \) |
| 29 | \( 1 + 7.80T + 29T^{2} \) |
| 31 | \( 1 - 2.75T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 - 0.193T + 41T^{2} \) |
| 43 | \( 1 + 5.67iT - 43T^{2} \) |
| 47 | \( 1 - 2.75iT - 47T^{2} \) |
| 53 | \( 1 + 10.8iT - 53T^{2} \) |
| 59 | \( 1 + 4.85T + 59T^{2} \) |
| 61 | \( 1 - 6.85T + 61T^{2} \) |
| 67 | \( 1 + 1.24iT - 67T^{2} \) |
| 71 | \( 1 + 2.75T + 71T^{2} \) |
| 73 | \( 1 + 4.23iT - 73T^{2} \) |
| 79 | \( 1 + 8.56T + 79T^{2} \) |
| 83 | \( 1 - 0.133iT - 83T^{2} \) |
| 89 | \( 1 - 5.61T + 89T^{2} \) |
| 97 | \( 1 - 7.24iT - 97T^{2} \) |
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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
−8.411721118428434053253878151216, −7.54266605097704084001826076658, −6.88102829081873141914332729522, −5.96301599622954239757438993454, −4.70374699561850029766653023903, −3.78888817661061832529700081077, −3.63026384621231595027956916338, −2.12963505941075567247049638619, −1.39298710667495261494377066939, −0.07793272493196978172763638318,
2.21980072028892335002947113820, 2.76601422863649332459537230057, 4.44071437367594941192106992217, 5.07159907913348778936118957880, 5.90827274735706739815522853979, 6.33809193863908058329415174759, 7.19666055898254301071798711879, 8.052135556422932264769527930210, 8.708954109255271192610161511515, 9.044434464481245544043054080816
